Box Box Box Box

I wonder if I will ever get back to something like weekly blogging instead of monthly. Anyway, time to do the solution to the box puzzle from last time.

First, we might as well use numbers instead of colours, because thats the sort of people we are. Let the colours be called 0, 1, and 2. In this case, "all three the same" and "all three different" is the same as "the three add to 0 mod 3".

This means the top box is negative the sum of the two boxes in the second row. The second row left box is negative the sum of the two left boxes in the third row, and the second row right box is negative the sum of the two right boxes in the third row, meaning that the top box is made up from the third row by adding the first box + 2*the second box + the third box. We can see quickly that Pascals triangle and the occasional minus sign is all we need.

The 9th row of Pascals triangle (calling the top row zero, which I do) is 1,9,36,84,126,126,84,36,9,1, so add the boxes on the bottom row with these coefficents mod 3 to get the box on the top. As it turns out, 9, 36, 84, and 126 are all multiples of 3, so they don't even matter, only the two end boxes contribute at all. Finally, there is a minus sign, as every other row gets one.

Thus the solution is, if the bottom two end boxes are the same colour, then the box on top will be that colour, if they are different colours, the box on top will be the third colour.