Number of solutions

I've said before that there are quadrillions of solutions to Einstein's Sudoku.  This wasn't an exaggeration.  How do we find out how many solutions there are?  Well, first let's look at the 4x4 game.

In the first cell, in the upper left, there are four different different possibilities: the red, blue, orange, or green house.  Given after we choose one of these (let's say the orange house), then there are three possibilities for the next cell (the red, blue, or green house).  And after we choose one of those, there are two possibilities for the next house, and then only one choice left for the last house.

If you work out all the possible orders of the houses, there are 4 x 3 x 2 x 1 possibilities, which totals up to 24.  4 x 3 x 2 x 1 can also be written as "4 factorial," or "4!"

For each of the 24 ways there are to configure the houses, there are also 24 ways to arrange the people, 24 ways to arrange the fruit, and 24 ways to arrange the drinks.  The total amount of possible solutions on the 4x4 grid is 24 x 24 x 24 x 24.  Or 4! x 4! x 4! x 4!.  Or 4!⁴.

This totals up to 331,776 different possible solutions, just for the 4x4 grid.

For the 3x3 grid, there are 3! x 3! x 3!, or 3!³ solutions.  This totals up to 6³ or 216 solutions.

For the 5x5 grid, there are 5! x 5! x 5! x 5! x 5!, or 5!⁵ solutions.  That totals up to 24,883,200,000 (over 24 billion) possible solutions.

And lastly, for the 6x6 grid, there are 6!6 solutions, which is 139,314,069,504,000,000, over 100 quadrillion.