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{Lattice Paths}

The Question

Given a square grid, how could you determine the number of possible paths from the top left of the grid to the bottom right, assuming you can only move to the right and down?

Example: a 2 by 2 grid has 6 possible paths 

Same Approach, Different View

(0,0)

(1,0)

(0,1)

(2,0)

(1,1)

(0,2)

(2,1)

(2,2)

(1,1)

(1,2)

(2,2)

(2,1)

(2,1)

(1,2)

(2,2)

(2,2)

(2,2)

(2,2)

(1,2)

Possible Solution

function paths2Bottom(x, y, max){
	if (x == max || y == max){
		return 1;
	}
	return paths2Bottom(x+1, y, max) + paths2Bottom(x, y+1, max);
}

• Recursive
• Only works for small grids
• Demo of it failing on big grids

Memoized Solution

var paths = {};
function memoizer(x, y, max){
	if(paths[[x,y,max]]){
		return paths[[x,y,max]];
	}
	else{
		return paths[[x,y,max]] = paths2Bottom(x, y, max);
	}
}
function paths2Bottom(x, y, max){
	if (x == max || y == max){
		return 1;
	}
	return memoizer(x+1, y, max) + memoizer(x, y+1, max);
}

• Still Recursive
• Works on bigger grids

Alternate Approach

When mapping the possible paths, you might see a pattern -- Pascal's Triangle!

We also know that when you hit the right or bottom edge of a grid, there is only one possible path.

Memoized Solution

var paths = {};
function memoizer(x, y, max){
	if(paths[[x,y,max]]){
		return paths[[x,y,max]];
	}
	else{
		return paths[[x,y,max]] = paths2Bottom(x, y, max);
	}
}
function paths2Bottom(x, y, max){
	if (x == max || y == max){
		return 1;
	}
	return memoizer(x+1, y, max) + memoizer(x, y+1, max);
}

• Still Recursive
• Works on bigger grids

Conclusion  

Solution on Repl.it

Source from Project Euler