Let S be the set of {(1,1), (1,−1), (−1,1), (1,0), (0,1)}-lattice paths which begin at (1,1), do not use the same vertex twice, and never touch either the x-axis or the y-axis. Let Sx,y be the set of paths in S which end at (x,y).For how many ordered pairs (x,y) subject to 1≤x,y≤31, is |Sx,y| a multiple of 3?
Details and assumptions
A lattice path is a path in the Cartesian plane between points with integer coordinates.
A step in a lattice path is a single move from one point with integer coordinates to another.
The size of the step from (x1,y1) to (x2,y2) is (x2−x1,y2−y1).
The length of a lattice path is the number of steps in the path.
For a set S={(xi,yi)}ki=1, an S-lattice path is a lattice path where every step has size which is a member of S.
To determine the number of ordered pairs (x, y) where |Sx,y| is a multiple of 3, we need to understand the properties of the given set S and analyze the lattice paths.
Given:
- The set S = {(1,1), (1,-1), (-1,1), (1,0), (0,1)} consists of different step sizes.
- The lattice paths start at (1,1), do not repeat any vertex, and never touch the x-axis or y-axis.
- We need to count the ordered pairs (x,y) subject to 1≤x,y≤31 for which the number of paths to reach (x,y) from (1,1) is divisible by 3.
To approach this problem, we can use the principle of inclusion-exclusion and dynamic programming.
1. Initialize a 31x31 grid (or an array) to keep track of the number of paths for each lattice point.
- Each cell (x, y) represents the number of paths to reach that point from (1,1).
2. Since the lattice paths cannot touch the x-axis or y-axis, we can set the values of the cells on both axes to zero.
- Set grid[x][0] = 0 for all 1≤x≤30
- Set grid[0][y] = 0 for all 1≤y≤30
3. For all other cells in the grid (x, y) where 1≤x,y≤31, calculate the number of paths using dynamic programming.
- Initialize grid[1][1] = 1 as the starting point.
- Iterate over each cell (x, y) in the grid:
- If (x, y) is not on the x-axis or y-axis, calculate:
- grid[x][y] = grid[x-1][y] + grid[x][y-1]
- This represents the number of paths to reach (x, y) from the neighboring cells above (x-1, y) and to the left (x, y-1).
- This step follows the principle of dynamic programming, where the number of paths to reach a point is the sum of paths to its adjacent points.
- Else, if (x, y) is on the x-axis or y-axis, set grid[x][y] = 0 (as mentioned earlier).
4. After populating the grid, iterate over each cell (x, y) in the grid, where 1≤x,y≤31.
- Count the number of cells where grid[x][y] % 3 = 0, and increment a counter variable.
5. Finally, the value of the counter variable represents the number of ordered pairs (x, y) for which |Sx,y| is a multiple of 3.
By following the above steps, you can programmatically solve the problem and obtain the desired count efficiently.