Consider the following one-person game:
A player starts with score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it $a$) and writes two positive integers (call them $b$ and $c$) such that $b + c = a$. The player then adds $b × c$ to her score. She repeats the step several times until she ends up with all $1$s on the whiteboard. Then the game is over, and the final score is calculated.
Example: At the first step, a player erases $20$ and writes $14$ and $6$, and gets a score of $14 × 6 = 84$. In the next step, she erases $14$, writes $9$ and $5$, and adds $9 × 5 = 45$ to her score. Her score is now $84 + 45 = 129$. In the next step, she may erase any of the remaining numbers on the whiteboard: $5$, $6$ or $9$. She continues until the game is over.
Alya and Bob play the game separately. Alya manages to get the highest possible final score. Bob, however, manages to get the lowest possible final score. What is the difference between Alya’s and Bob’s final scores?
I tried to "decompose" into a few numbers and I get the same scores. I am not sure how to prove the conjecture that any numbers will yield the same score no matter which path is taken.
from Hot Weekly Questions - Mathematics Stack Exchange
Lucius
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