idx int64 | task int64 | sample int64 | question string | student_prefix string | student_prefix_tokens int64 | teacher_text string | teacher_tokens_nemo int64 | teacher_prefill_tokens int64 | ce_target_tokens int64 | seam_shift int64 |
|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 6, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we have to find the number of distinct valid paths from (0,0,0) to (4,6,2) in 3D space, with certain movement rules and avoiding a specific forbidden point. The movement rules are a bit tricky because of the bitwise AND operations. Let me break it down.
First, let's ... | 4,096 | such that x' includes all bits of x. So, x' can be written as x | m, where m is any number (since x | m will include all bits of x plus possibly more bits). However, m can have any bits, but when we do x | m, we get x' which includes x's bits.
Therefore, x' = x | m, where m is any integer ≥ 0.
But x' must not equal ... | 1,024 | 4,335 | 1,021 | 0 |
1 | 0 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 6, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to count the number of distinct paths from (0, 0, 0) to (4, 6, 2) in 3D space, with certain movement rules and avoiding a forbidden point. Let me try to break this down step by step.
First, let me understand the movement rules. From any given point (x, y, z), I can move in three ways:
1. Move ... | 4,096 | such that each coordinate must be a superset of the bits of the previous coordinate in that axis, perhaps we can model this as a bitmask problem where each coordinate can only have bits added, not removed.
Wait, but if each coordinate can only have bits added (i.e., the value can only increase in terms of bits), then... | 1,024 | 4,350 | 1,013 | 0 |
2 | 1 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 0, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem step by step. So, the task is to find the number of distinct valid paths from (0, 0, 0) to (2, 0, 6) in 3D space, with certain movement rules and avoiding a forbidden point (2, 0, 0). The movements allowed are either changing the x-coordinate while keeping y and z the same... | 4,096 | we get there?
Alternatively, perhaps you can reach x=2 by being at x=0 and moving along the x-axis to x=2. But as we saw before, moving from x=0 to x=2 would require x'=2 to satisfy x & x' = x. Since x=0, 0 & 2 = 0, which equals x=0. Therefore, yes! So, moving from x=0 to x=2 is allowed, because 0 & 2 = 0.
Wait, tha... | 1,024 | 4,336 | 1,017 | 0 |
3 | 1 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 0, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to solve this problem where I have to count the number of distinct valid paths from (0, 0, 0) to (2, 0, 6) in 3D space, with certain movement rules and avoiding a forbidden point. Let me try to break this down step by step.
First, let's understand the movement rules. At each step, if I'm at (x,... | 4,096 | 0).
- The y-coordinate must remain 0 throughout.
Wait, that's an important point! Because, if we look at the target point (2, 0, 6), the y-coordinate is 0. The forbidden point is (2, 0, 0). So, if y-coordinate is involved, then perhaps the y-coordinate must stay at 0 in all valid paths. Because any movement in the y... | 1,024 | 4,353 | 1,007 | 0 |
4 | 2 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 1, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to solve this problem where I have to find the number of valid paths from (0, 0, 0) to (3, 1, 0) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me try to break it down step by step.
First, let me understand the movement rules. At each step, from a poi... | 4,096 | possible paths? Since the problem is about counting paths, but the state space is infinite, we need to find some patterns or constraints that can be used to limit the possible steps.
Alternatively, perhaps the problem is designed such that the movement can only be done by incrementing or decrementing each coordinate ... | 1,024 | 4,361 | 1,014 | 0 |
5 | 2 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 1, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (3, 1, 0) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me try to understand the problem step by step.
First, the movement types: at each step, you can move in three directions:
1. Move al... | 4,096 | coordinate (but different from it). Similarly for y and z.
But how does this affect the possible moves? Let's see.
Let's take the starting point: (0, 0, 0). At this point, x=0, y=0, z=0.
For movement type 1: moving along x-axis. We need to find x' such that x AND x' = x. So x=0, so x AND x' must equal 0. Since x is... | 1,024 | 4,367 | 1,011 | 0 |
6 | 3 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (6, 1, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct valid paths from (0, 0, 0) to (6, 1, 0) in 3D space, moving only along specific types of steps, and avoiding certain forbidden points. The forbidden points are (4,1,0), (6,0,0), (2,1,0), (0,1,0).
First, let me understand the mo... | 4,096 | we want to compute the number of ways to get to (x, y, z), then for movement along x-axis, we can come from any (x_prev, y, z) where x_prev AND x = x_prev. Wait, no. Because movement is from (x_prev, y, z) to (x, y, z) if x_prev AND x = x_prev. So, to come to (x, y, z) via x-axis movement, the previous point must have... | 1,024 | 4,361 | 1,021 | 0 |
7 | 3 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (6, 1, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out this problem. We need to count the number of distinct valid paths from (0,0,0) to (6,1,0) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific here. Let me break it down.
First, the movement types:
1. Moving along the x-axis: (x', y, z) such that ... | 4,096 | 1, 0).
So the forbidden points have either y=1 (and z=0 except (6,0,0)), or y=0 (like (6,0,0)).
Now, the approach is to compute for each point (x, y, z) the number of ways to reach it from (0,0,0), without going through any forbidden points.
Since the forbidden points are known, we can set dp[x][y][z] = 0 for any fo... | 1,024 | 4,356 | 1,023 | 0 |
8 | 4 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 3, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's see. I need to find the number of distinct paths from (0, 0, 0) to (4, 3, 5) in 3D space, moving only in specific ways. The allowed moves are:
1. Move along the x-coordinate: (x', y, z) where x AND x' = x. So, x' must have the same bits as x, but maybe with some bits set? Wait, no. Wait, the AND op... | 4,096 | 4 (100), 5 (101), 6 (110), 7 (111), 12 (1100), etc. So any number that has the third bit set.
But wait, x AND x' = x would be true if x' has all bits of x. So, for x=4 (binary 100), x' must have the third bit set. So x' can be 4, 5, 6, 7, 12, 13, 14, 15, etc.
But the previous x' (the x-coordinate of the state we cam... | 1,024 | 4,357 | 1,018 | 0 |
9 | 4 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 3, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (4, 3, 5) in 3D space, avoiding three specific forbidden points. The movement rules are a bit tricky. Let me break it down.
First, the allowed moves are:
1. Move along the x-coordinate, keeping y and z the same. So, (x, y, z) can mov... | 4,096 | predecessors for (x,y,z) along the x-axis, it's the number of x' such that (x', y, z) can move to (x, y, z). So, the move from (x', y, z) to (x, y, z) is allowed if x' AND x = x. Wait, no. Let's re-examine.
The move is from (x', y, z) to (x, y, z) if (x', y, z) is a type 1 move: x AND x' = x. So, x AND x' = x. So, x ... | 1,024 | 4,361 | 1,024 | 0 |
10 | 5 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (6, 6, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0,0,0) to (6,6,4) in 3D space, moving in specific ways, avoiding certain forbidden points. The movement rules are a bit tricky. Let me break it down step by step.
First, the movement types:
1. Moving along the x-axi... | 4,096 | First, let's list all possible x, y, z values. Since we are moving from (0,0,0) to (6,6,4), the maximum x is 6, maximum y is 6, maximum z is 4. So x ranges from 0 to 6, y from 0 to 6, z from 0 to 4.
Forbidden points are given as a list. Let's list all forbidden points:
(0, 6, 0)
(0, 2, 0)
(6, 0, 0)
(2, 4, 4)
(4, 6, 4... | 1,024 | 4,469 | 1,023 | 0 |
11 | 5 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (6, 6, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (6,6,4) in 3D space, avoiding certain forbidden points. The movement allowed is changing one coordinate at a time, but with specific constraints. Let me try to break this down.
First, let me understand the movement types:
1. Moving in ... | 4,096 | 2^k where k is any bit not set in x. Wait, but in x=3, bits 0 and 1 are set. So any additional bits would be in higher positions. For example, x' = 3 | 4 = 7 (111). So x AND x' = 3 AND 7 = 3, which is correct. Similarly, x'=3 | 8=11 (but that's beyond 6). So for x=3, possible x' are 3 (same), 7 (but too big), etc. Wai... | 1,024 | 4,456 | 1,022 | 0 |
12 | 6 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 0, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we have to find the number of distinct paths from (0, 0, 0) to (1, 0, 1) in 3D space, moving only in certain ways, and avoiding the point (0, 0, 1). The movement rules are a bit tricky with the AND operations. Let me break it down step by step.
First, let me understa... | 4,096 | can jump to any x' > 0. But for x>0, you can only move to x' ≥ x.
But since the target for x is 1, maybe we don't need to consider x higher than 1? Let me check.
Similarly for the z-coordinate: the target is (1, 0, 1), so z=1. So, perhaps for z, we need to see how we can get from z=0 to z=1.
Let's think about z-coo... | 1,024 | 4,342 | 1,010 | 0 |
13 | 6 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 0, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (1, 0, 1) in 3D space, moving according to specific rules, and avoiding the forbidden point (0, 0, 1). The answer needs to be modulo 4377.
First, let me make sure I understand the movement rules correctl... | 4,096 | there are infinite possibilities. However, the problem is asking for paths to (1, 0, 1), so perhaps only certain x', y', z' values are relevant.
Wait, perhaps the problem is that the movement should only be to points where all coordinates are non-negative? Or maybe the problem expects that you can only move to points... | 1,024 | 4,333 | 1,024 | 0 |
14 | 7 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 6, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (2,6,2) in 3D space, with certain movement rules and avoiding specific forbidden points. Let me try to break this down step by step.
First, let me understand the movement rules. At each step, from a point (x, y, z), I can move in three ... | 4,096 | allowed moves and find a way to count the paths.
But given that the grid is infinite (since coordinates can be any integers), but the forbidden points are specific, and the target is (2,6,2), perhaps we can limit our consideration to some small grid around the target.
But perhaps there's a way to model this using dy... | 1,024 | 4,351 | 1,017 | 0 |
15 | 7 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 6, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0,0,0) to (2,6,2) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me try to break it down.
First, the movement types are:
1. Moving in the x-direction: (x', y, z) where x AND x' = x. Wait, that's a bit confus... | 4,096 | (x, y, z) in increasing order (maybe x from 0 to 2, y from 0 to 6, z from 0 to 2), but we need to make sure that when we process (x, y, z), all possible previous states have already been processed.
But given that the transitions can go from lower x to higher x, etc., maybe it's better to process states in order of in... | 1,024 | 4,359 | 1,013 | 0 |
16 | 8 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 3, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (1, 3, 3) without going through any of the forbidden points. The movement allowed is in three directions: along the x-axis, y-axis, or z-axis, but each step must flip the bit in that coordinate. Wait, actually, the movement types are:
... | 4,096 | example, maybe the movement is like moving in a binary reflected Gray code, but I'm not sure.
Let me try to think of small examples. Let's take the starting point (0,0,0). From here, what moves can we make?
In x-direction: moving to (x', 0, 0) where x' can be any integer except 0. Because x is 0, so 0 AND x' = 0, wh... | 1,024 | 4,431 | 1,020 | 0 |
17 | 8 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 3, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (1,3,3) in 3D space, avoiding certain forbidden points. The movement allowed is changing one coordinate at a time, but with specific conditions. Let me break this down step by step.
First, the allowed moves are:
1. Changing x-coordinate... | 4,096 | , y, or z directions. That makes sense. Because once you set a bit in x, you can't unset it. Because to move from x to x', you need x AND x' = x, which implies that x' has at least the bits of x. Therefore, x is a non-decreasing sequence. Similarly for y and z.
Therefore, the entire path is such that x, y, z are non-d... | 1,024 | 4,411 | 1,022 | 0 |
18 | 9 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 1, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's tackle this problem step by step. So, the task is to find the number of distinct valid paths from (0,0,0) to (2,1,2) in 3D space, avoiding the forbidden point (0,1,2). The allowed moves are to change one coordinate in a specific way: for each coordinate, you can only move to a point where that coord... | 4,096 | is allowed. Therefore, moving from (0,0,0) to (2,0,0) is allowed, because x=0 to x=2 is a type 1 move (moving along x-coordinate).
Similarly, once at (2,0,0), x is now 2. So, if you want to move along the x-coordinate again, you need x AND x' = 2. So x' must be such that 2 AND x' = 2. So, x' must have the bits set th... | 1,024 | 4,346 | 1,016 | 0 |
19 | 9 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 1, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0, 0, 0) to (2, 1, 2) in 3D space, moving only through specific types of steps, and avoiding the forbidden point (0, 1, 2). The movement rules are:
1. Move to a point with the same y and z coordinates, but different x. So, (x', y, z) where x AND x' = x... | 4,096 | 0, and we are considering x' as part of the x-coordinate (which can only be 0,1,2), so x' can be 1 or 2. So for x=0, possible x' are 1,2.
If x=1 (binary 001), then x' must satisfy x' AND 1 = 1. So x' must have the least significant bit set. So possible x' values are 1, 3, 5, etc. But given x' must be in {0,1,2}, so po... | 1,024 | 4,338 | 1,000 | 0 |
20 | 10 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 5, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem step by step. So, we need to count the number of distinct valid paths from (0, 0, 0) to (4, 5, 2) in 3D space, with specific movement rules and avoiding a forbidden point. The movement rules are a bit tricky, so I need to understand them properly.
First, let's parse the m... | 4,096 | ways to reach (x, y', z) for all y' such that y & y' = y and y' != y.
- The number of ways to reach (x, y, z') for all z' such that z & z' = z and z' != z.
But this seems complicated, because for each (x, y, z), we need to sum over all possible x' that satisfy the condition.
However, since the x, y, z coordinates ar... | 1,024 | 4,335 | 1,020 | 0 |
21 | 10 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 5, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0, 0, 0) to (4, 5, 2) in 3D space, where each step is moving in one of three directions: along the x-axis, y-axis, or z-axis, but with the condition that the bit AND operation must be maintained. Also, I can't visit the point (4, 1, 2). The answer needs... | 4,096 | all bits unset), then moving along x-axis can set any bits (since x' can be anything, as 0 AND x' = 0). Wait, but 0 has no bits set, so x' can be anything. Therefore, moving along x-axis from x=0 can set any bits.
If x is, say, 1 (binary 001), then moving along x-axis would require x' to have the LSB set. So x' can be... | 1,024 | 4,325 | 1,019 | 0 |
22 | 11 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 4, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's tackle this problem. So, the task is to find the number of distinct paths from (0, 0, 0) to (1, 4, 4) in 3D space, moving only in certain ways and avoiding three forbidden points. The movement is allowed in three types: moving along the x-axis (same y and z, but x changes), moving along the y-axis (... | 4,096 | as a subset (which it always does). Therefore, from any (x, 0, z), you can move along y to any y' (except 0) by moving along type 2.
Similarly for z.
So, the movement is similar across all axes. The key is that for each axis, when the current value is 0, you can move to any non-zero value along that axis. When the c... | 1,024 | 4,353 | 1,011 | 0 |
23 | 11 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 4, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0, 0, 0) to (1, 4, 4) in 3D space, avoiding three specific forbidden points: (1,4,0), (0,4,4), and (1,0,4). The movement allowed is to change one coordinate at a time, but with some constraints. Let me break down the problem step by step.
First, let me... | 4,096 | that has a bit not set in the original x. Wait, no, the condition is x AND x' = x, which means that x' must have all the bits that x has. So x can't lose any bits when moving. Therefore, x' must be a superset of x. So the movement in type 1 allows you to add bits to x, but you can't remove bits. Similarly for type 2 a... | 1,024 | 4,361 | 1,017 | 0 |
24 | 12 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 3, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, I need to find the number of distinct valid paths from (0, 0, 0) to (3, 3, 6) in 3D space, avoiding the forbidden point (0, 3, 2). The movement rules are a bit specific. Let me break it down.
First, let's understand the movement types. From any point (x, y, z), you can move:
1. To (x', y, z) where x AND... | 4,096 | and only if x' is a superset of the bits of x. Which implies that x' >= x in terms of the bits set.
Wait, not exactly. For example, x=2 (binary 10), x' can be 2, 3 (10 | 01 = 11), 6 (110), etc. So x' must be >= x in the sense that when you set the bits of x, x' can have additional bits.
But perhaps another way to th... | 1,024 | 4,339 | 1,021 | 0 |
25 | 12 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 3, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (3,3,6) in 3D space, avoiding the point (0,3,2). The movement rules are a bit specific. Let me try to parse this.
First, the movement types:
1. Move to (x', y, z) where x AND x' = x. So, x' must have the same bits as x, right? Wait, no... | 4,096 | along x-axis from (x, y, z) can only go to (x', y, z) where x' has x as a subset of bits. Similarly for y and z.
Alternatively, perhaps we can think of the movement as being allowed only to increase the coordinates in certain ways, but I'm not sure.
Alternatively, perhaps the movement rules imply that the path must ... | 1,024 | 4,332 | 1,023 | 0 |
26 | 13 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 3, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out how to solve this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (2, 3, 0) in 3D space, moving according to specific rules, avoiding certain forbidden points. The forbidden points are (2,2,0), (2,0,0), (2,1,0), (0,1,0), (0,3,0), (0,2,0).
First, I need... | 4,096 | , then that path would be invalid.
Wait, but this seems to complicate things. Let me think again. Perhaps I've misunderstood the movement rule. Maybe it's supposed to be (x', y, z) such that x AND x' = 0? But the problem states x AND x' = x. Alternatively, maybe it's a typo and it's supposed to be x AND x' = 0? But as... | 1,024 | 4,369 | 1,019 | 0 |
27 | 13 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 3, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem step by step. So, we need to find the number of distinct valid paths from (0, 0, 0) to (2, 3, 0) in 3D space, following certain movement rules and avoiding specific forbidden points. The movement rules are a bit tricky, so I need to understand them carefully.
First, let's... | 4,096 | a non-zero z, and then back to 0? But moving back to 0 would require that from z', we can move to z=0 such that z' AND 0 = z'. Wait, but z' AND 0 = 0. So z' AND 0 must equal z', which implies that z' must be 0. But z' is not 0 (since we moved away from it). Therefore, moving along z-axis away from 0 and then back to 0... | 1,024 | 4,374 | 1,018 | 0 |
28 | 14 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 1, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So we need to count the number of distinct paths from (0,0,0) to (3,1,4) in 3D space, avoiding certain forbidden points. Each step can be in one of three directions: moving along the x-axis (same y and z), y-axis (same x and z), or z-axis (same x and y). Also, we can't vi... | 4,096 | Let's think about what moves are possible from a given point (x, y, z).
For example, from (x, y, z), moving along x-axis: can go to any (x', y, z) where x AND x' = x (and x' != x). Similarly for y and z.
But since the problem requires us to count paths, perhaps dynamic programming is still feasible if we can correct... | 1,024 | 4,418 | 1,023 | 0 |
29 | 14 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 1, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (3, 1, 4) in 3D space, moving only in specific directions, and avoiding certain forbidden points. The movement rules are a bit specific. Let me break it down step by step.
First, let me understand the mov... | 4,096 | of x.
Therefore, moving from (x', y, z) to (x, y, z) is allowed if x' has all bits of x (i.e., x' is a superset of x in binary).
But the movement direction is from (x', y, z) to (x, y, z) is possible if x' is a superset of x.
Wait, but for example, if x is 0, then x AND x' = 0. So, moving from (x', y, z) to (0, y, ... | 1,024 | 4,431 | 1,016 | 0 |
30 | 15 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 6, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to figure out the number of distinct valid paths from (0, 0, 0) to (1, 6, 1) in 3D space, avoiding certain forbidden points. Let me start by understanding the movement rules and then think about how to model this problem.
First, the movement types:
1. Move along the x-axis: (x', y, z) where x ... | 4,096 | is a multiple of x (but not exactly; it's numbers where x is a subset of x' in terms of bits set? Or the other way around?).
Wait, perhaps it's better to think of the movement as being able to move along the x-axis only if you can set or unset bits in x, but only in such a way that you don't unset any bits that are a... | 1,024 | 4,395 | 1,023 | 0 |
31 | 15 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 6, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (1, 6, 1) in 3D space, avoiding certain forbidden points. The movement rules are a bit tricky. Let me think step by step.
First, let's understand the movement rules. At each step, from a point (x, y, z), you can move in three directio... | 4,096 | you can move to other x's. But since we are trying to reach (1,6,1), which has x=1, maybe moving in x-direction can only take us between x=0 and x=1. Let's think:
If we are at x=0, moving to x' where x AND x' = x=0. So x' can be any value. So, moving from x=0, you can go to x'=0, but you can't stay. But you can go to... | 1,024 | 4,411 | 1,014 | 0 |
32 | 16 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 2, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of valid paths from (0, 0, 0) to (3, 2, 0) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me break it down.
First, the movement types:
1. Moving along the x-axis: from (x, y, z) to (x', y, z) where x AND x' = x. Wait, but x AND x' ... | 4,096 | to reach (3,2,0). So (3,2,0) is allowed; it's the endpoint.
But forbidden points are those six listed, which do not include (3,2,0). So (3,2,0) is allowed.
But the forbidden points are all in z=0. So the forbidden points are points where z=0, x and y have certain values.
So, for example, (0,2,0) is forbidden, but (... | 1,024 | 4,383 | 1,021 | 0 |
33 | 16 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 2, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (3, 2, 0) in 3D space, with certain movement rules and avoiding specific forbidden points. Let's break it down step by step.
First, let's understand the movement rules. At each step, you can move in t... | 4,096 | are 1, 3, 5, 7, etc. But since x can be at most 3 (since we need to reach x=3), the possible supersets for x=1 (binary 01) are 1 and 3 (binary 11).
Wait, x can be up to 3, so x' can be at most 3 as well? Wait, no. Because for example, if x is 1, x' can be 3 (binary 11), which is allowed. Because in the problem statem... | 1,024 | 4,371 | 1,018 | 0 |
34 | 17 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 4, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to figure out the number of valid paths from (0,0,0) to (4,4,3) in 3D space, avoiding specific forbidden points. The movement rules are a bit tricky. Let me break it down.
First, the movement types:
1. Moving along the x-axis: (x', y, z) where x AND x' = x. Wait, that's a bit confusing. Let me... | 4,096 | is 2 (10), then x'=1 (01) would have x AND x' = 0, which is not equal to x (10). So moving to x'=1 is not allowed from x=2. So from x=2, you can't move to x'=1. So in this case, moving to a lower x is not allowed. Similarly, moving from x=3 (11) to x'=1 (01) would result in x AND x' = 1, which is not equal to x (3), s... | 1,024 | 4,384 | 1,010 | 0 |
35 | 17 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 4, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's tackle this problem. So, the task is to find the number of distinct valid paths from (0, 0, 0) to (4, 4, 3) in 3D space, with certain movement rules and avoiding specific forbidden points. The movement rules are a bit tricky, so I need to understand them properly.
First, let's parse the movement op... | 4,096 | to a different point, so x' != 4.
Similarly, if x=0, then x' can be anything (since 0 AND x' = 0), so you can move to any x' != 0.
But this is a bit confusing. However, maybe for the purpose of this problem, since the target is (4,4,3), we can consider that we only need to explore coordinates up to (4,4,3) (or perha... | 1,024 | 4,380 | 1,020 | 0 |
36 | 18 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 6, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0,0,0) to (4,6,1) in 3D space, moving in specific ways, avoiding certain forbidden points. The movement rules are a bit tricky. Let me try to parse them again.
First, the allowed moves:
1. Move to (x', y, z) where x... | 4,096 | . So, binary 0001 (assuming 3-bit numbers, but since x can be up to 4, which is 100, but maybe leading zeros are okay). Wait, the problem doesn't specify bit-length, but since x is up to 4, which is 3 bits (100), perhaps we can assume numbers are represented with enough bits to cover up to 4. So, for x=1 (binary 001), ... | 1,024 | 4,360 | 1,023 | 0 |
37 | 18 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 6, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. It's about finding the number of distinct paths from (0, 0, 0) to (4, 6, 1) in 3D space with certain movement rules and avoiding forbidden points. The movement rules are a bit tricky, so I need to understand them properly.
First, let's parse the movement types:
1. Movin... | 4,096 | , if x=4 (100), then x' can be 4, 5 (101), 6 (110), 7 (111), 8 (1000), etc. But since the target point is (4,6,1), maybe x can only go up to 4? Because if x' exceeds 4, then you can't get back to 4. For example, if you go to x=5, then to get to x=4, you would need to move from x'=5 to x=4, but 4 & 5 = 4? Let's see: 4 i... | 1,024 | 4,367 | 1,013 | 0 |
38 | 19 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 3, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to solve this problem where I have to count the number of distinct paths from (0, 0, 0) to (3, 3, 0) in 3D space, with certain movement rules and avoiding forbidden points. Let me try to break this down step by step.
First, let me understand the movement rules. The allowed moves are:
1. Move a... | 4,096 | 0 or 1. Therefore, for x AND x' to be x, x' must have all bits set that x has, and can have more bits set. Therefore, x' must be a superset of x. Therefore, x' can be any number that is a superset of x's bits. In that case, x AND x' = x. Let's test this.
Take x=3 (binary 11). Then x' can be any number that has bits s... | 1,024 | 4,398 | 1,018 | 0 |
39 | 19 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 3, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to figure out the number of distinct valid paths from (0, 0, 0) to (3, 3, 0) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me break it down.
First, the movement types:
1. Move along the x-axis: (x', y, z) where x AND x' = x. Wait, that's a bit confu... | 4,096 | ), you cannot make any moves? Because moving along x, y, or z axes would require x', y', z' to be different but with the AND condition. Since x=0, moving along x is impossible (as x' must be 0). Same for y and z. Therefore, you cannot move at all from (0,0,0)? That would make the problem impossible. But the problem say... | 1,024 | 4,410 | 1,014 | 0 |
40 | 20 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 0, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to figure out the number of distinct paths from (0, 0, 0) to (5, 0, 0) in 3D space, moving only along specific types of steps, and avoiding two forbidden points: (1, 0, 0) and (4, 0, 0). The allowed moves are:
1. Move in the x-direction: (x', y, z) where x AND x' = x. Wait, that's a bit confusi... | 4,096 | figure out possible x-values that can be reached.
But let's consider x first. Let's list the possible x-values that can be part of a path from 0 to 5.
Starting at x=0.
From x=0, moving in x-direction: since x=0, x AND x' = 0 for any x'. So, possible x' are any numbers except 0 (since x' must be different from x). S... | 1,024 | 4,337 | 1,021 | 0 |
41 | 20 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 0, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So we need to find the number of distinct paths from (0,0,0) to (5,0,0) in 3D space, moving only in certain ways, avoiding two specific points. The movement rules are a bit tricky because of the bitwise AND operations. Let me try to break it down step by step.
First, let... | 4,096 | set.
Therefore, for each x, the x' is any number such that x OR x' = x' (i.e., x is a subset of x' in terms of bits). So, x' >= x in terms of bits. But numerically, x' can be larger than x, or have the same value? But since x AND x' = x, x' must have all bits of x set. So numerically, x' can be any number that is gre... | 1,024 | 4,340 | 1,024 | 0 |
42 | 21 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 3, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's see. I need to find the number of distinct valid paths from (0, 0, 0) to (4, 3, 6) in 3D space, avoiding certain forbidden points. The movement rules are a bit tricky. Let me try to break it down.
First, the movement types. From any point (x, y, z), you can move in three directions:
1. (x', y, z) ... | 4,096 | x' values.
Let me try to list possible x' for each x (x from 0 to 4):
x: 0:
possible x' where 0 AND x' = 0 → x' can be any (since 0 AND anything is 0). So x' ∈ {0,1,2,3,4}, but x' ≠ 0 → x' ∈ {1,2,3,4}
x: 1:
x is 001. x AND x' must equal x → x' must have the 0th bit set. So x' can be 1 (001), 3 (011), 5 (101), 7 (11... | 1,024 | 4,369 | 1,011 | 0 |
43 | 21 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 3, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (4, 3, 6) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me try to break this down.
First, the movement types:
1. Move along the x-axis: (x', y, z) where x AND x' = x. Wait, that's a bit co... | 4,096 | Move along x-axis to (x', y, z) where x' is any value except x, but only if x=0 (since otherwise x AND x' =x only holds when x'=x).
Wait, no. Let me rephrase:
Movement along x-axis is allowed only if:
If current x is 0: you can move to any x' ≠0.
If current x is non-zero: you can move only to x'=x (but that's same... | 1,024 | 4,407 | 1,018 | 0 |
44 | 22 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 6, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (5,6,3) in 3D space with certain movement rules, avoiding specific forbidden points. Let me try to break this down step by step.
First, let's understand the movement rules. At each step, you can move in three directions:
1. Move along ... | 4,096 | predecessor x' must have x' AND x = x, but x' ≠ x.
Thus, for each (x, y, z), the predecessors are:
- All (x', y, z) where x' AND x = x and x' ≠ x
- All (x, y', z) where y' AND y = y and y' ≠ y
- All (x, y, z') where z' AND z = z and z' ≠ z
Then, dp[x][y][z] = sum of dp[x'][y][z] for valid x' + sum of dp[x][y'][z]... | 1,024 | 4,424 | 1,018 | 0 |
45 | 22 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 6, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct valid paths from (0,0,0) to (5,6,3) in 3D space, moving through points by changing one coordinate at a time, but avoiding certain forbidden points. The movement rules are a bit tricky because of the bitwise AND operations. Let me... | 4,096 | ][0] = 1.
Then, for each other point, we compute dp[x][y][z] as sum of transitions from possible previous states.
But since this is a DP problem where transitions can come from any previous state (not just smaller coordinates), we need to process the states in the correct order.
Wait, but since you can move to any x... | 1,024 | 4,429 | 1,022 | 0 |
46 | 23 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 6, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (5, 6, 1) avoiding certain forbidden points. The allowed moves are changing one coordinate at a time, but with some conditions. Let me try to break this down.
First, the movement rules. From any point (x, y, z), I can move to:
1. (x'... | 4,096 | ≠ 5. Wait, no. Wait, to reach (5, 6, 1), the predecessor in x-direction would be a point (x', 6, 1) such that x' AND 5 = x'. Because when moving from (x', 6, 1) to (5, 6, 1), the condition is that x' AND 5 = x'. Wait, no. The movement is from (x', y, z) to (x, y, z) where x' AND x = x'. So when moving from x' to x, x'... | 1,024 | 4,492 | 1,013 | 0 |
47 | 23 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 6, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, the task is to find the number of distinct valid paths from (0, 0, 0) to (5, 6, 1) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific: at each step, you can move in three directions: along the x-axis, y-axis, or z-axis, but only if t... | 4,096 | Wait, but this seems like you can only move along an axis if you are already at a point with a certain bit pattern. So, to increase or decrease the x-coordinate, you can only move to x' that has the same bits as x.
Wait, but this is confusing. Let's think differently. Perhaps the movement in the x-direction is allowed... | 1,024 | 4,475 | 1,022 | 0 |
48 | 24 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 5, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to solve this problem where I have to find the number of distinct paths from (0, 0, 0) to (1, 5, 0) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me try to break it down step by step.
First, let's understand the movement options. At any point (x, y, ... | 4,096 | 0) (y=0, z=0). So, if z is 0, then some points are forbidden.
But given that the target is z=0, maybe the path ends at z=0. So, perhaps we can model the problem by considering that z is allowed to be any value, but we need to reach z=0 at the end.
But this seems complex. However, given that the forbidden points inclu... | 1,024 | 4,370 | 1,014 | 0 |
49 | 24 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 5, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (1, 5, 0) in 3D space, moving through certain steps, avoiding specific forbidden points. The movement rules are a bit tricky with the bitwise AND operations. Let me break this down step by step.
First, le... | 4,096 | have to reach z=0 at the target. But the forbidden points have z=0 (all of them), so z=0 is allowed except for the forbidden points.
Wait, the forbidden points are (0,4,0), (1,4,0), (1,1,0), (0,5,0), (1,0,0). All of them have z=0. So, all forbidden points are in the z=0 plane. So, perhaps we can consider the entire p... | 1,024 | 4,395 | 1,015 | 0 |
50 | 25 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 0, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (3, 0, 1) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific: each step can be moving in one of three directions:
1. Along the x-axis: (x', y, z) where x AND x' = x. So, x' must have the same bits as... | 4,096 | in the positions where x has 1s. Therefore, x' can be any number where those bits are set. However, the AND of x and x' will be x only if x' has all the bits that x has. Therefore, for x=3, x' must have bits 0 and 1 set (assuming 0-based from the right). Therefore, x' can be 3 (0011), 5 (0101), 6 (0110), 7 (0111), etc... | 1,024 | 4,373 | 1,011 | 0 |
51 | 25 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 0, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to count the number of distinct paths from (0, 0, 0) to (3, 0, 1) in 3D space, with certain movement rules and avoiding specific forbidden points. The movement rules are a bit tricky, so I need to understand them properly.
First, let me restate the movement t... | 4,096 | be any number where bits 0 is set. So x' can be 1, 3, 5, etc., but since x' != x_curr, x' can be 3,5,7,... However, since the target x is 3, maybe we only need to consider up to x=3. Wait, but maybe the movement can go beyond x=3? For example, if x_curr is 1, you could move to x=5 (binary 101), but then how would you ... | 1,024 | 4,364 | 1,021 | 0 |
52 | 26 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 1, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. It's a bit complex, but I'll take it step by step. The goal is to find the number of distinct valid paths from (0,0,0) to (4,1,6) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific, so I need to understand them properly.
First, let me r... | 4,096 | maybe higher bits? But since x can't exceed 4, perhaps x' can only be numbers up to 4.
Wait, the x-coordinate is between 0 and 4. So for x in 0-4, the possible x' values are in 0-4 as well.
So, for each x (0<=x<=4), find all x' in 0<=x'<=4, x' !=x, such that x & x' = x.
Let's tabulate these.
First, let's consider ... | 1,024 | 4,451 | 1,004 | 0 |
53 | 26 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 1, 6). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (4, 1, 6) in 3D space, avoiding certain forbidden points. Each step can be moving along one of three directions: changing x, y, or z, but with the condition that the new coordinate must have the same bits as the original in the respect... | 4,096 | allowed to any x' (different from current), so x' can be any integer except 0. However, since our target is (4, 1, 6), perhaps we can limit x' to be between 0 and 4. Because moving beyond 4 would not help in reaching the target. But maybe the problem allows moving to any x', but since we need to end at (4,1,6), any ex... | 1,024 | 4,462 | 1,017 | 0 |
54 | 27 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 5, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out how to solve this problem. So, we need to count the number of distinct valid paths from (0, 0, 0) to (1, 5, 5) in 3D space, avoiding certain forbidden points. The movement is allowed in three types: moving along the x-axis, y-axis, or z-axis, but with the condition that each move m... | 4,096 | x'=3, x'=5, etc. But if you are at x=3, then x' must have the same bits as 3 (binary 11), so x' can be 3, 7, etc. So, moving to higher x's is possible, but moving back to lower x's may not be possible. For example, from x=3, can you move to x=1? Let's see: x=3 is binary 11, x'=1 is binary 01. Then x AND x' = 3 & 1 = 1... | 1,024 | 4,487 | 1,018 | 0 |
55 | 27 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 5, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out how to solve this problem. So, we need to count the number of distinct valid paths from (0, 0, 0) to (1, 5, 5) in a 3D space, avoiding certain forbidden points. The movement rules are a bit specific, involving AND operations, and I need to make sure I understand all the constraints... | 4,096 | ensure that none of the intermediate steps are forbidden.
But the problem is that the space is infinite, unless the coordinates can be bounded. However, the forbidden points are all given and have certain coordinates, and the target is (1,5,5). Therefore, perhaps the necessary coordinates are only up to certain maxim... | 1,024 | 4,489 | 1,022 | 0 |
56 | 28 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 1, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. It seems a bit complex, but I'll break it down step by step.
First, the problem is about finding the number of distinct paths from (0, 0, 0) to (2, 1, 1) in 3D space, with certain movement rules and avoiding specific forbidden points. The movement rules are a bit tricky... | 4,096 | because 0 AND 2 = 0. So from (0, y, z), you can move to any (x', y, z) where x' is any value (since 0 AND x' = 0) except 0. But given that the end point is x=2, maybe we only need to consider x up to 2, y up to 1, z up to 1. Because if we go beyond, for example, x=3, then it might not help us reach (2,1,1). Similarly ... | 1,024 | 4,351 | 1,019 | 0 |
57 | 28 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 1, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct valid paths from (0, 0, 0) to (2, 1, 1) in 3D space, moving in specific ways, avoiding certain forbidden points. The movement rules are a bit tricky, so I need to understand them properly.
First, let me parse the movement types:... | 4,096 | ,1,1), perhaps we don't need to consider x beyond 2? Because if we move to x=3 or higher, we can't get back? Wait, but maybe you can. For example, if x=2, which is 10 in binary, then moving along x-axis would require x' such that 2 AND x' = 2. So x' can be 2, 3 (11), 6 (110), etc. But if you move to x=3, then from ther... | 1,024 | 4,357 | 1,022 | 0 |
58 | 29 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 3, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (1, 3, 0) in 3D space, avoiding certain forbidden points. The allowed moves are in three directions: moving along the x, y, or z axis, but with specific conditions. Let me first try to understand the problem thoroughly.
First, the mov... | 4,096 | ' can be anything, so moving to x=1 is allowed. Wait, but x=0 to x=1 is allowed as long as the movement is along x-axis, and x' is not equal to x (so x'=1 is different from 0, so allowed). But (1,0,0) is forbidden, so moving from (0,0,0) to (1,0,0) is not allowed because it's forbidden.
But how else can you get to x=1... | 1,024 | 4,397 | 1,008 | 0 |
59 | 29 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 3, 0). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (1, 3, 0) in 3D space, avoiding certain forbidden points. Let me try to break this down.
First, let me understand the movement rules. From any point (x, y, z), you can move to:
1. (x', y, z) where x AND x' = x. Wait, that seems a bit... | 4,096 | ): allowed.
(0, 2, 0): forbidden.
(0, 3, 0): forbidden.
(1, 0, 0): forbidden.
(1, 1, 0): forbidden.
(1, 2, 0): forbidden.
(1, 3, 0): destination.
So, the allowed states are:
(0, 0, 0): start.
(0, 1, 0).
(1, 3, 0): destination.
Wait, but wait, maybe there are other allowed states? Let's check:
For x=0:
- y=... | 1,024 | 4,375 | 1,023 | 0 |
60 | 30 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 6, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (4,6,2) in 3D space, avoiding certain forbidden points. Let me try to break this down.
First, the movement rules. At each step, you can move in one of three directions:
1. Move along the x-axis: (x', y, z) where x AND x' = x. Wait, but... | 4,096 | (x + delta_x, y, z), where delta_x is any number such that (x + delta_x) & x = x. But I'm not sure if that's helpful.
Alternatively, perhaps the movement along x-axis is equivalent to moving to x' = x | k, where k is any number (non-zero?) such that x | k ≠ x. Wait, no. Because x | k will have all bits of x plus any ... | 1,024 | 4,447 | 1,017 | 0 |
61 | 30 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 6, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct valid paths from (0, 0, 0) to (4, 6, 2) in 3D space, considering the movement rules and avoiding certain forbidden points. The movement types are:
1. Move along the x-axis: (x', y, z) where x AND x' = x. Wait, that seems a bit c... | 4,096 | or maybe 5 is beyond the allowed x.
Wait, but in the problem statement, the target is (4, 6, 2), so the x ranges up to 4. So, in our grid, x can be from 0 to 4 inclusive. Similarly for y (0 to 6) and z (0 to 2).
So, for each x, we can generate all possible x' such that x' is between 0 and max_x (4) and x' ≠ x and x ... | 1,024 | 4,473 | 1,019 | 0 |
62 | 31 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 0, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to solve this problem where I have to count the number of distinct paths from (0,0,0) to (2,0,4) in 3D space, with certain movement rules and avoiding a specific forbidden point. Let me try to break this down.
First, let me understand the movement rules. At each step, from a point (x,y,z), I ca... | 4,096 | to y=0 is not allowed along y-axis. So once you leave y=0 via movement along y-axis, you can't return to y=0. Therefore, the only way to have y=0 is if you never move along y-axis. Similarly for x and z axes.
Wait, but the target point has y=0, so the only way to reach the target is to not move along y-axis at all. B... | 1,024 | 4,344 | 1,018 | 0 |
63 | 31 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 0, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0, 0, 0) to (2, 0, 4) in 3D space, moving only in specific ways. The movement rules are a bit tricky, and there's a forbidden point I need to avoid. Let me try to break this down step by step.
First, let's understand the movement types:
1. Moving in t... | 4,096 | be 2, 6, 10, 14, etc. So, essentially, x' is x | k for some k ≥ 0. Wait, no. Because x | k would be the OR of x and k. If x is 2 (10), then x | k is 10 | k. So, for example, if k is 0100 (4), then x | k is 110 (6). If k is 0000, then x | k is 2. So, x' can be any number obtained by setting some bits that are not set i... | 1,024 | 4,346 | 1,020 | 0 |
64 | 32 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 5, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem step by step. First, I need to understand what exactly the problem is asking for.
We have a 3D space starting at (0, 0, 0) and need to get to (0, 5, 5). The allowed moves are:
1. Move along the x-coordinate: (x', y, z) where x AND x' = x. Wait, so if we're at (x, y, z), ... | 4,096 | for that axis.
This seems a bit tricky. Let me try to think of possible paths.
We start at (0, 0, 0). From here, we can move along x-axis (since x=0), y-axis (since y=0), or z-axis (since z=0). So from (0,0,0), we can move to:
- Any (x', 0, 0) where x' ≠ 0 (x-axis moves).
- Any (0, y', 0) where y' ≠ 0 (y-axis move... | 1,024 | 4,463 | 1,007 | 0 |
65 | 32 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 5, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. Hmm, so we need to find the number of distinct valid paths from (0,0,0) to (0,5,5) in 3D space, moving only in specific ways, and avoiding certain forbidden points. The movement rules are a bit tricky with the bitwise AND operations. Let me start by understanding the move... | 4,096 | along y:
y can be 1 (forbidden), 2,3,4,5 (but 5 is forbidden as (0,5,0)), so y=2,3,4 → 3 possibilities.
Similarly, z can be 1 (forbidden), 2,3,4,5. But (0,0,5) is forbidden, so z=2,3,4 → 3 possibilities.
So total of 6 possible initial moves.
Now, but each of those moves leads to a point, and from there, we can mak... | 1,024 | 4,447 | 1,017 | 0 |
66 | 33 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 1, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (4,1,5) in 3D space, following certain movement rules and avoiding specific forbidden points. Let me try to break this down step by step.
First, let me understand the movement rules. At each step, from a point (x, y, z), you can move in... | 4,096 | 0, then y AND y' = 0 for any y', so movement type 2 allows moving to any y' ≠ y. Similarly for z.
Oh! That makes much more sense. So my previous misunderstanding was that I thought x AND x' = x implies x' must have x's bits. But actually, it's the other way around. If x AND x' = x, then x must be a subset of the bits... | 1,024 | 4,366 | 1,004 | 0 |
67 | 33 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 1, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (4, 1, 5) in 3D space, moving only through certain types of steps. Each step can either change one of the coordinates x, y, or z, but with some conditions. Also, we can't visit certain forbidden points. Th... | 4,096 | to x'=1, 2, 3, 4, etc., but since x cannot exceed 4 (as target is 4), perhaps x' can be up to 4.
Wait, but the movement rules don't limit x', y', z' to be within any range. However, since we are moving towards (4,1,5), and forbidden points are within certain ranges, perhaps we can restrict our DP to x in [0,4], y in ... | 1,024 | 4,365 | 1,017 | 0 |
68 | 34 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 3, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (4, 3, 5) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific: at each step, you can move along x, y, or z coordinates, but only in a way that the bitwise AND with the current coordinate remains the sa... | 4,096 | x as a subset in terms of bits.
However, if x is 0 (all bits 0), then x AND x' = 0 for any x'. Therefore, the condition x AND x' = x (which is 0) is satisfied for any x'. Therefore, moving along x-coordinate when x=0 is allowed to any x' (different from current x, which is 0). So, x' can be any non-zero number.
This... | 1,024 | 4,506 | 1,020 | 0 |
69 | 34 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (4, 3, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out how to solve this problem. So, the task is to count the number of distinct valid paths from (0,0,0) to (4,3,5) in 3D space, avoiding certain forbidden points. The movement is allowed in specific ways: you can move along the x-axis, y-axis, or z-axis, but only if the new coordinate ... | 4,096 | 4, the only x' is 4.
Wait, but when moving from x' to x, if x is 4, then x' must be a superset of x. So, x' can't be less than x (in terms of bits). So, for example, if x is 0, then x' can be any number, because x' AND 0 = 0. So, to reach (x, y, z) via moving along x-axis, the previous x' can be any number (from 0-4) ... | 1,024 | 4,471 | 1,021 | 0 |
70 | 35 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (6, 4, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's tackle this problem step by step. So, we need to find the number of distinct paths from (0, 0, 0) to (6, 4, 4) in 3D space, moving in certain directions, while avoiding the forbidden point (0, 4, 4). The allowed moves are:
1. Move in the x-direction: (x', y, z) where x AND x' = x. Wait, that seems ... | 4,096 | ), perhaps we can model this using dynamic programming.
Let's consider that the coordinates x, y, z are non-negative integers. Because the starting point is (0,0,0) and the target is (6,4,4). Also, moving in x-direction may increase or decrease x? Wait, no. Wait, moving in x-direction: the x-coordinate must satisfy x ... | 1,024 | 4,338 | 1,020 | 0 |
71 | 35 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (6, 4, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (6, 4, 4) in 3D space, moving in specific ways, while avoiding the forbidden point (0, 4, 4). The allowed moves are:
1. Move along the x-axis: (x', y, z) where x AND x' = x. So, x' must have the same bits... | 4,096 | fact that moving along x can generate multiple steps? Wait, no: each step is a single move of one of the three types.
Thus, for the DP transition, from (x, y, z), you can move to any of the next positions allowed by each type, and add the current count to those positions.
But since the allowed next positions are man... | 1,024 | 4,330 | 1,023 | 0 |
72 | 36 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 2, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (2,2,4) in 3D space, avoiding the forbidden point (2,0,4). Each step can be moving in one of three directions: changing x, y, or z, but with the condition that the bitwise AND of the new coordinate with the current one equals the origina... | 4,096 | , and x' ≠ x.
But since x is small (0,1,2), we can enumerate all possible x' for each x.
Let's enumerate possible transitions for x:
Case 1: x = 0.
x AND x' = 0 → always true. So x' can be any number except 0. But since x can be up to 2, the possible x' values are 1, 2.
Case 2: x = 1.
x is 1 (binary 01). So x ... | 1,024 | 4,379 | 1,006 | 0 |
73 | 36 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 2, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0, 0, 0) to (2, 2, 4) in 3D space, avoiding the forbidden point (2, 0, 4). The movement allowed is to change one coordinate at a time, but with specific constraints based on the bitwise AND operation. Let me try to break this down.
First, let's underst... | 4,096 | Define the DP table dp[x][y][z], representing the number of ways to reach (x, y, z) without visiting any forbidden points so far.
2. Initialize dp[0][0][0] = 1.
3. For all (x, y, z), if it's the forbidden point (2, 0, 4), set dp[x][y][z] = 0 and do not process it.
4. For each (x, y, z), compute the number of ways t... | 1,024 | 4,340 | 1,012 | 0 |
74 | 37 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 1, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, the task is to find the number of distinct valid paths from (0, 0, 0) to (1, 1, 3) in a 3D space, moving according to specific rules, while avoiding five forbidden points. And the answer needs to be modulo 8465.
First, let's understand the movement rules. At each st... | 4,096 | , 1), (0, 0, 2), etc. But we need to get to (1, 1, 3). So, points like (0,1,0) are part of the forbidden points? Let me check the forbidden points: (1,1,1), (1,1,0), (0,1,0), (1,0,0), (1,0,1). So, (0,1,0) is forbidden, as is (1,0,0). So, moving from (0,0,0) to (0,1,0) is invalid because (0,1,0) is a forbidden point. Si... | 1,024 | 4,371 | 1,013 | 0 |
75 | 37 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 1, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct valid paths from (0, 0, 0) to (1, 1, 3) in 3D space, moving according to specific rules, and avoiding certain forbidden points. The forbidden points are (1,1,1), (1,1,0), (0,1,0), (1,0,0), (1,0,1). And each move has to be of one ... | 4,096 | the x and y coordinates can only be 0 or 1. Because if x is more than 1, say x=2, then moving to x=1 would be allowed? Let's check.
For example, if you are at x=2 (binary 10), then moving to x'=1 (binary 01) would require that 1 & 2 = 2? No. 1 & 2 is 0, which is not equal to 2. So, moving from x=2 to x=1 is not allow... | 1,024 | 4,395 | 1,019 | 0 |
76 | 38 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 2, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (0,2,5) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me try to understand the problem step by step.
First, the movement types:
1. Move to (x', y, z) where x AND x' = x. So, x' has to have t... | 4,096 | 0,y',0) where y'≠0, and any (0,0,z') where z'≠0.
Now, let's think about movement from a point like (x, 0, 0) where x≠0.
From (x, 0, 0), we can move along the x-axis: x AND x' = x, which requires x' to have all the bits of x. Since x≠0, x' must be a superset of x's bits. So, x' can be any number that includes x's bits... | 1,024 | 4,353 | 1,004 | 0 |
77 | 38 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 2, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (0, 2, 5) in 3D space, avoiding three specific forbidden points. The allowed moves are moving in the x, y, or z directions, but with specific constraints. Let me try to break this down.
First, let me understand the movement rules. At ... | 4,096 | can't do that. Similarly for others.
But perhaps, the x-coordinate is fixed at 0 throughout, because the movement along x is not possible if we want to stay at x=0. Let's test this.
If we are at (0, y, z), then moving in type 1 (x-direction) would require x' such that x AND x' = x (x=0) so x' can be any number, but ... | 1,024 | 4,355 | 1,020 | 0 |
78 | 39 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 1, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (1, 1, 3) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific: at each step, you can move along the x-axis, y-axis, or z-axis, but only to points where the bitwise AND with the current coordinate remai... | 4,096 | (1,1,3). Let's check if it's forbidden. The forbidden list: yes, (1,1,3) is not listed as forbidden. So (1,1,3) is allowed.
So allowed nodes are:
(0,0,0), (1,1,3). And maybe some others? Let me check again.
Forbidden list:
Looking at all 16 nodes:
x=0, y=0:
z=0: allowed (start)
z=1: (0,0,1) → forbidden.
z=2: (... | 1,024 | 4,447 | 1,022 | 0 |
79 | 39 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 1, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. It's about finding the number of valid paths from (0,0,0) to (1,1,3) in 3D space with certain movement rules and avoiding forbidden points. Hmm.
First, let me understand the movement rules. At each step, you can move in three directions:
1. Along the x-axis: (x', y, z) ... | 4,096 | 1,1,3). The forbidden points include points like (1,0,3), (0,1,2), etc.
Wait a minute, maybe the coordinates are limited to 0 and 1 for x and y, and 0 to 3 for z. Because the target is (1,1,3). So perhaps the x and y coordinates can only be 0 or 1? Because the forbidden points include (1,0,3), etc., which have x and y... | 1,024 | 4,446 | 1,017 | 0 |
80 | 40 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 0, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (5, 0, 2) in 3D space, with certain movement rules and avoiding specific forbidden points. Let me break down the problem step by step.
First, let me understand the movement rules. At each step, you can mo... | 4,096 | For example, from x=1 (binary 01), you can move to x=3 (binary 11) or x=5 (binary 101), etc. But x=3 and x=5 are within 0 to 5. However, from x=5, which is 101 in binary, moving along x-axis would allow moving to x' where x' & 5 = 5. So, x' can be 5, 7 (111), etc. But since the target is at x=5, perhaps moving beyond ... | 1,024 | 4,397 | 1,013 | 0 |
81 | 40 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 0, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (5,0,2) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific: at each step, you can move in one of three directions:
1. Along the x-axis: (x', y, z) where x AND x' = x. So, the x-coordinate can stay the ... | 4,096 | x' = x and x' != x.
Similarly, from (x, y, z), you can move along y-axis to (x, y', z) where y & y' = y and y' != y.
Similarly for z-axis.
But since we're doing DP forward, it's perhaps easier to think: for each (x, y, z), we can look at all possible previous positions (x_prev, y_prev, z_prev) that can move to (x, ... | 1,024 | 4,374 | 1,025 | 0 |
82 | 41 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 2, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (3, 2, 4) in 3D space, moving only along specific directions and avoiding certain forbidden points. Let me try to break this down step by step.
First, let me understand the movement rules. At each step, from a point (x, y, z), I can m... | 4,096 | , let me think again. For x=1 (binary 1), to move to x' such that 1 AND x' = 1. So x' must have the 0th bit set. So x' can be 1 (binary 1), 3 (binary 11), 5 (101), etc. So from x=1, you can move to x'=1 (not allowed), 3, 5, 7, etc. So x' must be >=1 and have the 0th bit set. So moving to higher numbers.
Similarly, for... | 1,024 | 4,433 | 1,020 | 0 |
83 | 41 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 2, 4). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (3, 2, 4) in 3D space, avoiding certain forbidden points. The movement allowed is to change one coordinate at a time, but with specific conditions. Let me try to break this down.
First, the movement types:
1. Move in the x-direction:... | 4,096 | 4 (7), etc. Wait, so x' can be 3, 7, 11, etc. So for example, if x is 3 (11), then x' can be 3, 7 (111), 11 (1011), etc. So numbers like 3, 7, 15, etc., which have the lower two bits set.
Therefore, moving in x-direction when x is 3 would allow you to move to x' values like 3, 7, 11, 15, etc. But the only way to move... | 1,024 | 4,419 | 1,017 | 0 |
84 | 42 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 5, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem step by step. It's a path-finding problem in 3D space with some constraints. Let me break down what's given and figure out how to approach it.
We start at (0, 0, 0) and need to reach (2, 5, 3). The allowed moves are:
1. Move along the x-coordinate: from (x, y, z) to (x',... | 4,096 | to any numbers. However, given that the forbidden points are all in the range where x is 0 or 2, y is 0,1,4,5, etc., and z is 0,1,2,3, perhaps we can assume that all possible x, y, z in the path are limited to a certain range.
Wait, but for example, if x can be any number, but if you have to reach (2,5,3), then the x... | 1,024 | 4,403 | 1,018 | 0 |
85 | 42 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 5, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (2, 5, 3) in 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me break it down.
First, the allowed moves are:
1. Moving along the x-coordinate: from (x, y, z) to (x', y, z) where x AND x' = x. W... | 4,096 | ?
Wait, the target is (2,5,3). So, x goes from 0 to 2, y from 0 to 5, z from 0 to 3.
But according to the movement rules, x can only increase. So, for example, you can't have x decrease from 2 to 1. So, to get to x=2, you need to have the x-coordinate reach 2 exactly when you move into it. Since you start at x=0, the... | 1,024 | 4,440 | 1,009 | 0 |
86 | 43 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 2, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem step by step. So we have to find the number of distinct valid paths from (0, 0, 0) to (0, 2, 1) in 3D space, with certain movement rules and avoiding two forbidden points. The movement rules are a bit tricky, but let's break them down.
First, the movement types:
1. Move ... | 4,096 | 0, then x' can be any number, because 0 AND x' = 0, which equals x (which is 0). Wait, this is the key! Oh! Wait a second. If the condition is x AND x' = x, then:
x AND x' = x implies that x is a subset of x' in terms of bits. Because if x has a bit set, then x' must have that bit set. If x has a bit not set (0), the... | 1,024 | 4,355 | 1,023 | 0 |
87 | 43 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 2, 1). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. It's a bit complex, but I'll take it step by step.
First, the problem is about finding the number of distinct paths from (0, 0, 0) to (0, 2, 1) in 3D space, with certain movement rules and avoiding forbidden points. The movement rules are:
1. Moving in the x-direction: ... | 4,096 | z might not need to be too large. For example, perhaps we can restrict ourselves to a certain range.
Let's think: the target is (0,2,1). Let's see what possible values x, y, z can take.
For x: starting at 0, and whenever we move along x-axis, x' must have same bits as x. Since we start at x=0, any x' is allowed (exc... | 1,024 | 4,343 | 1,024 | 0 |
88 | 44 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 2, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct paths from (0, 0, 0) to (1, 2, 3) in 3D space, following certain movement rules and avoiding specific forbidden points. Let me try to break this down step by step.
First, let's understand the movement rules. At each step, I can move in three ways:
1. Move along t... | 4,096 | 2, 3) where x' AND 1 = 1 (so x' must be odd), x' ≠ 1; plus all (1, y', 3) where y' AND 2 = 2 (so y' must have the bits set in 2, which is binary 10, so y' must be even and have the second bit set? Wait, 2 in binary is 10. So, y' AND 2 = 2 implies that the second bit (bit 1) of y' must be set. So, y' can be 2, 3, 6, 7,... | 1,024 | 4,351 | 1,019 | 0 |
89 | 44 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (1, 2, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (1, 2, 3) in 3D space, avoiding three specific forbidden points. The movement allowed is moving along x, y, or z coordinates in such a way that each move keeps the same bit in the coordinate. Let me try to break this down.
First, let ... | 4,096 | is equal to the sum over:
- All possible x' such that x' AND x = x (and x' ≠ x), of dp[x'][y][z]
- All possible y' such that y' AND y = y (and y' ≠ y), of dp[x][y'][z]
- All possible z' such that z' AND z = z (and z' ≠ z), of dp[x][y][z']
But this seems like an infinite sum because x', y', z' can be any numbers. H... | 1,024 | 4,358 | 1,017 | 0 |
90 | 45 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 4, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (2, 4, 5) in 3D space, avoiding the point (0, 4, 5). The movement is allowed in three types: moving along the x-axis (keeping y and z the same, but changing x such that x AND x' = x), moving along the y-axis (changing y and keeping x a... | 4,096 | & y' = y. So, similar to the x-axis case. Therefore, moving along y or z axis can only increase the respective coordinate.
Wait, but if the starting point is (0,0,0), and the target is (2,4,5), then all coordinates need to increase. So, all moves are along the positive x, y, or z direction. So, each move increases on... | 1,024 | 4,341 | 1,017 | 0 |
91 | 45 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 4, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct valid paths from (0, 0, 0) to (2, 4, 5) in 3D space, moving only along specific types of steps and avoiding the forbidden point (0,4,5). The movement rules are a bit tricky with the bitwise AND operations. Let me break it down.
... | 4,096 | I got the condition reversed. Perhaps it's x' AND x = x, but that's the same as x AND x' = x. Alternatively, maybe it's x' AND x = x', but that would mean x' is a subset of x.
Alternatively, maybe it's x' AND x = x', which would mean x' is a subset of x. Let's check.
If the movement was (x', y, z) where x' AND x = x... | 1,024 | 4,325 | 1,018 | 0 |
92 | 46 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 3, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to figure out the number of valid paths from (0, 0, 0) to (2, 3, 3) in 3D space, avoiding certain forbidden points. Each step can be moving in one of three directions: changing the x-coordinate while keeping y and z the same, changing the y-coordinate while keeping x and z the same, or changing ... | 4,096 | can't exceed some maximum? Wait, no. The problem doesn't state any upper limit on the coordinates except that the forbidden points are given. So theoretically, the x-coordinate can go higher than 2, but since the forbidden points are given, some of those points may be forbidden, but others may be allowed. However, the... | 1,024 | 4,479 | 1,004 | 0 |
93 | 46 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (2, 3, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out this problem. So, we need to find the number of distinct paths from (0,0,0) to (2,3,3) in 3D space, moving in specific ways, avoiding certain forbidden points. The movement rules are a bit tricky. Let me break it down.
First, the allowed moves are:
1. Moving along the x-axis: (x'... | 4,096 | predecessors along the x-direction are all (x', y, z) where x' has x & x' = x and x' ≠ x.
Similarly for y and z.
But in order to compute the DP, we need to find for each (x, y, z), which (x', y, z), (x, y', z), (x, y, z') are allowed predecessors.
But since the movement can go both ways (i.e., x can increase or dec... | 1,024 | 4,495 | 1,019 | 0 |
94 | 47 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 3, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (3, 3, 2) in a 3D space, avoiding certain forbidden points. The movement rules are a bit specific. Let me break it down.
First, the movement rules: at each step, you can move along one of three types of steps:
1. (x', y, z) where x A... | 4,096 | perhaps it's easier to think in terms of the numbers directly.
So, x AND x' = x => x' must have all the bits that x has, but may have additional bits. So, for example, if x=1 (0001), x' can be 1 (0001), 3 (0011), 5 (0101), etc. But since x can be up to 3, x' for x=1 is 1 and 3. Because 5 is higher than 3, but in our ... | 1,024 | 4,418 | 1,022 | 0 |
95 | 47 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (3, 3, 2). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0,0,0) to (3,3,2) avoiding certain forbidden points. The allowed moves are to any of the three types mentioned: moving along the x-coordinate (keeping y and z the same), moving along the y-coordinate (keeping x and z the same), or moving along the... | 4,096 | x, and x & x' = x.
Similarly for type 2 and type 3.
Therefore, for each (x, y, z), we can generate:
- For type 1: for each x' in possible_x_moves(x) (i.e., x' != x and x & x' = x), the point (x', y, z).
- For type 2: for each y' in possible_y_moves(y) (y' != y and y & y' = y), the point (x, y', z).
- For type 3: ... | 1,024 | 4,402 | 1,020 | 0 |
96 | 48 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 0, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. It's a three-dimensional path counting problem with some restrictions. Let me start by understanding what exactly needs to be done.
So, we have to find the number of distinct valid paths from (0, 0, 0) to (5, 0, 3) using specific movement rules, avoiding certain forbidde... | 4,096 | 0,1,0) is forbidden. But the forbidden points are given as (5,0,2), etc. So only certain points with y=0 are forbidden.
Wait, looking at the forbidden points: they all have y=0. Because (5,0,2), (1,0,2), (1,0,1), (0,0,2), (5,0,0). All of them have y=0. So the forbidden points are all in the plane y=0.
Therefore, any ... | 1,024 | 4,381 | 1,017 | 0 |
97 | 48 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (5, 0, 3). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to figure out this problem. We need to count the number of distinct paths from (0,0,0) to (5,0,3) in 3D space, moving according to specific rules, and avoiding certain forbidden points. The movement rules are a bit tricky, so I need to break them down.
First, let me understand the movement type... | 4,096 | , y, z) to (x, y', z), then y can change. However, since the target y is 0, it's possible that you might have to go through some y values and come back to 0.
But given that the movement in y is allowed when you are at a certain y, you can move to any y' (if current y is 0) or to any y' that has the bits set in current... | 1,024 | 4,390 | 1,013 | 0 |
98 | 49 | 0 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 3, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, let's try to tackle this problem. So, we need to find the number of distinct paths from (0, 0, 0) to (0, 3, 5) in 3D space, moving only through specific types of steps, avoiding certain forbidden points. The allowed moves are:
1. Moving in the x-direction: (x', y, z) such that x AND x' = x. Wait, that's ... | 4,096 | must be a misunderstanding of the movement rules.
Let me re-examine the problem statement.
"At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | 1,024 | 4,405 | 1,024 | 0 |
99 | 49 | 1 | In a three-dimensional space, you start at point (0, 0, 0) and want to reach the point (0, 3, 5). At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) point of one of the following types:
1. (x', y, z) such that x AND x' = x
2. (x, y', z) such that y AND y' = y
3. (x, ... | <think>
Okay, so I need to find the number of distinct valid paths from (0, 0, 0) to (0, 3, 5) avoiding certain forbidden points. Let me break down the problem step by step.
First, the movement rules. At each step, from a point (x, y, z), I can move in three directions:
1. Move along the x-coordinate: (x', y, z) wher... | 4,096 | you can't move anywhere from (0,0,0)? This would mean that there are no paths, but that's probably not the case. So I must be misunderstanding the movement rules.
Wait, let me check the problem statement again:
"At each step, if you are currently at (x, y, z), you may move to a new (different from the current one) p... | 1,024 | 4,386 | 1,019 | 0 |
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