The Math This section of Puzzle Help contains a few thoughts about the math behind the puzzle. The Number of Combinations Think about this sequence: When you start out, the first cell you fill can have any one of nine digits. The second cell you fill can have any one of the remaining eight digits. The third cell you fill can have any one of the remaining seven digits. And so on... When you get to the last cell, there's only one digit remaining, so it's not much of a choice. Do you see the pattern? The total number of combinations can be easily calculated: 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 This pattern occurs frequently. Mathematicians call it a "factorial", They even have a special notation, using an exclamation point. So they would say that the number of combinations in this puzzle is 9!, pronounced "9 factorial". Did you try multiplying those numbers? Go ahead, get out your calculator and do it. You'll find that 9! is equal to 362,880. That's a lot of combinations, so it's not surprising that the puzzle is difficult to solve. The Solutions In the Solution tab, run or skip to a solution, and then try again repeatedly. You will find that the program finds four solutions. Skip back and forth between them and examine the results. The four solutions are reflections of each other on the horizontal and vertical axes. So there is really just a single solution, flipped around the four possible different ways. Another way of saying this is that the arrangement of the cells is symmetrical on the horizontal and vertical axes, so the solutions must have the same symmetry.