A game of removing stones can look like a long, boring subtraction. But if you look at what never changes, a hidden measure appears and governs the whole process.
The Two Piles
Riddle statement
There are two piles of stones.
One has 187 stones and the other has 121.
On each move, subtract the smaller pile from the larger pile. Repeat the process as long as the piles are different.
How many stones will each pile have at the end?
Show solution
Solution
Both piles will end with 11 stones.
Apply the rule:
From there, the pile of 55 keeps losing 11 stones at each step until it also becomes 11.
The deeper reason is that subtracting one number from the other does not change their common divisors. Therefore, the final value is the greatest common divisor of the two initial numbers.
Also, 187 and 121 are divisible by 11:
and they do not share a larger divisor.
Answer: both piles end with 11 stones.