# Sunday Times Teaser 3131 – Heads Up Savings

*by Howard Williams*

#### Published Sunday September 25 2022 (link)

Little Spencer saves 5p coins in a jar, and when they reach £10, deposits them in his savings account. He likes playing with the coins. In one game, after first turning them all heads up, he places them in a row on the table. Starting from the left, he then turns over every 2nd coin until he has reached the end of the row. He then again starts from the left, and this time turns over every 3rd coin. He repeats this for every 4th, 5th coin etc, until finally he turned over just one coin, the last in the row.

At the end of the game I could see that if Spencer had exactly 75 per cent more coins he would have an increase of 40 per cent in the number showing heads. However, if he had exactly 50 per cent fewer coins, he would have a decrease of 40 per cent in the number showing heads.

What is the value of his 5p savings?

If we number the coins from 1 upwards, a coin at position N (> 1) will be turned over once for each divisor of N greater than 1. And, since non-square / square numbers have odd / even numbers of divisors, the game will finish with only the coins at square positions showing heads. This hence provides a faster way of counting the number of heads showing after each game.

Similar.

Doing the 50% decrease check first seems to be more efficient.

Based on Brian’s observation that square numbers have an even number of divisors and non square numbers have an odd number of divisors…