# Sunday Times Teaser 3222 – Squarism

*by Mark Valentine*

#### Published Friday June 21 2024 (link)

There are 84 marbles, numbered sequentially from 1, in a bag. Oliver (going first) and Pam take turns, each removing two marbles from the bag. The player finds the “most square” integer factorisation of each number. The difference between the two factors for each number is then added to the player’s score (eg removing marbles 1 and 84 scores 5 points since 1=1×1 and 84=7×12), and the game ends when all marbles are removed.

After each of the first four turns (two each), both players’ expected final score was a whole number. For example, if there were 90 points remaining, with 4 turns left for Oliver and 5 for Pam, Oliver would expect to score another 40 points. In addition, each player’s score for their second turn was the same as their first. Also, Pamela scored the lowest possible game total.

What was Oliver’s expected final score after Pamela’s second turn?

A lot faster but more complex: