Every split produces a different number and opens new possibilities. The total seems to depend on the chosen path—until you discover what each product is really counting.
The Divided Loot
Riddle statement
You begin with 10 tokens in a single pile.
At each step:
- Choose a pile containing at least two tokens.
- Split it into two nonempty piles.
- Record the product of the sizes of the two new piles.
Repeat the process until all 10 tokens are separated, one in each pile.
Then add all the recorded products.
What sum do you obtain? Can it depend on how you make the splits?
What is the result if you start with n tokens?
Show solution
Solution
Answer: the sum is always 45. It does not depend on how you make the splits.
With \(n\) tokens, the general result is:
Why
Consider each pair of tokens. Initially, the two tokens in a pair are together. At the end, they lie in different piles.
When a pile is split into parts of sizes \(a\) and \(b\), exactly the pairs with one token on each side become separated. There are:
such pairs, which is precisely the product recorded at that step.
Every pair of tokens becomes separated for the first time in exactly one step. Therefore, adding all the recorded products counts every pair exactly once.
The total is the number of pairs that can be formed from \(n\) tokens:
For \(n=10\):
Alternative check
Let \(E\) be the sum of \(\binom{s}{2}\) over the sizes of all current piles. A split \(a+b\to a,b\) decreases \(E\) by exactly \(ab\). Since \(E\) starts at \(\binom{n}{2}\) and ends at \(0\), the sum of all recorded products must be \(\binom{n}{2}\).