Here is a generalized solution to the [path-counting brain teaser from Kahn Academy](https://www.khanacademy.org/math/math-for-fun-and-glory/puzzles/brain-teasers/v/3-d-path-counting-brain-teaser). You can run this in node passing the size and number of dimensions as arguments.

path-count.js

// argv: size, dims
'strict mode';

var size = process.argv[2];
var dims = process.argv[3];
var grid = initializeGrid(size, dims);

setValueAt(createArray(dims, 0), grid, 1);

var initialLoc = createArray(dims, size-1);
countPaths(initialLoc);

console.log(JSON.stringify(grid));
console.log(getValueAt(createArray(dims, size-1), grid));

/**
 * Initializes the n-cube grid. The result is an array of arrays n-deep.
 * 
 * @param  {number} s size of the n-cube
 * @param  {number} d number of dimensions of the n-cube.
 * 1 is a line, 2 is a square, 3 is a cube, etc.
 * 
 * @return {Array} the n-cude with all entries initialized to 0
 */
function initializeGrid(s, d) {
	if (d <= 1) {
		return createArray(size, null);
	}

	var line = [];
	for (var j = 0; j < s; j++) {
		line.push(initializeGrid(s, d - 1));
	}
	return line;
}

function countPaths(loc) {
	var neighbors = getNeighbors(loc);
	var sums = neighbors.map(neighbor => {
		// use memoized value if possible
		var value = getValueAt(neighbor, grid);
		if (value === null) {
			value = countPaths(neighbor);
		}
		return value;
	});
	myValue = sums.reduce((acc, pathCount) => acc + pathCount, 0);
	setValueAt(loc, grid, myValue);
	return myValue;	
}

function getNeighbors(loc) {
	var neighbors = [];
	for (var i = 0; i < dims; i++) {
		if (loc[i] > 0) {
			var neighbor = copyLoc(loc);
			neighbor[i]--;
			neighbors.push(neighbor);
		}
	}
	return neighbors;
}

function getValueAt(loc, gridPiece) {
	if (loc.length === 1) {
		return gridPiece[loc[0]];
	}
	return getValueAt(loc.slice(1), gridPiece[loc[0]]);
}

function setValueAt(loc, gridPiece, value) {
	if (loc.length === 1) {
		gridPiece[loc[0]] = value;
		return;
	}
	setValueAt(loc.slice(1), gridPiece[loc[0]], value);
}

function copyLoc(loc) {
	return JSON.parse(JSON.stringify(loc));
}

/**
 * Creates an array of the given size with all elements set to val.
 * 
 * @param  number length The length of the array
 * @param  any val    The value to set all elements to.
 * @return Array<any>   The array
 */
function createArray(length, val) {
	var array = [];
	array.length = length;
	array.fill(val);
	return array;
}