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?