##### Problem 15

# Lattice paths

Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.

How many such routes are there through a 20×20 grid?

I initially started to do the brute force calculation. From top-left to bottom-right step by step ( grid size * 2 ) but that is brutal on the PC even on step 20.

Then there is this solution starting from the back right-bottom to top-left each cell has two points to move backwards.

Solution:

`var gridSize = 20; var pathSize = gridSize * 2; // due to the rules of only right and down movement var grid = {};`

for( var cell = 0; cell < gridSize; cell++ ) { grid[cell+"x"+gridSize] = 1; grid[gridSize+"x"+cell] = 1; }

for( var row = gridSize-1; row >= 0; row-- ) { for( var cell = gridSize-1; cell >= 0; cell-- ) { grid[row+'x'+cell] = grid[(row+1)+"x"+cell] + grid[row+"x"+(cell+1)]; } }

`console.log( grid['0x0'] );`