Othello Puzzle Solution

Firstly, I would like to thank everyone who attempted my first puzzle I hope you enjoyed it. Here is the solution to the puzzle.

As you can see the number that have their white side facing up are: 1 , 4 , 9 , 16 ,  25 , 36 , 49 , and 64. These are the square numbers. If you continued this puzzle on for ever, only the square numbers would act this way. Why do the square numbers behave in this way? The key to the solution of this puzzle is the title of my blog itself. This puzzle is all about factors. Usually the factors of a number come in pairs. For example the number 24, has the factors: 1 , 2 , 3 , 4 , 6 , 8 , 12 and 24. These factors can be paired into two numbers that multiply together to make 24. This gives you: 1 and 24 ,  2 and 12 ,  3 and 8 and finally 4 and 6. As these factors can be paired, they have an even number of factors. However, square numbers have an odd number of factors. If we take the number 36 and repeat the process, the result is different. We pair, 1 and 36 ,  2 and 18 ,  3 and 12 , 4 and 9 and finally 6 and 6. Here the six is repeated so we only count it once. This is the reason that the square numbers have an odd number of factors. My next puzzle will be uploaded very soon!