At first sight, the puzzle seems to be rather easy. Looking at the first circles on the top: 99 and 72 pointing to 27, it makes sense to assume the following pattern: 92 – 72 = 27. The same pattern continues following the next few circles down the line: 45 – 27 = 18 and 39 – 18 = 21. Which means that the missing number must be the difference between 36 and 21, which is 15.
For completeness’s sake, we follow this rule down the tree, and it works, till OOPs, 21 – 13 = 7.
What has gone wrong??? There is nothing wrong with the puzzle. We simply should have applied a different, less obvious rule. Let me explain.
All the digits from any two circles pointing simultaneously to a third circle, summed up, produce the resulting number on that circle. For example, summing up 72 and 99 in this way gives 7 + 2 + 9 + 9 resulting in 27, the number on the circle to which the arrows point. Using this technique, the resulting number on the circle with the question mark is 12, i.e. 2 + 1 + 3 + 6 = 12.
This puzzle is brilliant because of its uniqueness. Yoshigahara has found two arithmetical rules that can be applied to the same numbers for five steps in a row, but only one of which fails for the final step, and then only by one.