## Teaser 2869: Cubic savings

**From The Sunday Times, 17th September 2017** [link]

In 2009 George and Martha had a four-figure number of pounds in a special savings account (interest being paid into a separate current account). At the end of the year they decided to give some of it away, the gift being shared equally among their seven grandchildren, with each grandchild getting a whole number of pounds. At the end of the following year they did a similar thing with a different-sized gift, but again each grandchild received an equal whole number of pounds. They have repeated this procedure at the end of every year since.

The surprising thing is that, at all times, the number of pounds in the savings account has been a perfect cube.

What is the largest single gift received by any grandchild?

[teaser2869]

## Jim Randell 10:54 am

on7 May 2020 Permalink |This puzzle is marked as flawed, as there are two possible solutions.

I assumed the number of grandchildren remained constant (at 7) during the eight years in question (2009 – 2016).

The amounts in the savings account are perfect cubes, that differ by multiples of 7, so we can collect cubes by their residue modulo 7, and consider the sets for each residue to look for 9 amounts that satisfy the remaining conditions.

This Python program runs in 76ms.

Run:[ @repl.it ]Solution:There are two solutions. The largest amount is received by a grandchild is £ 292, or £ 388.If we start with 5832 (= 18³) in the savings account, and then give presents of (248, 103, 292, 86, 31, 64, 8, 1) to each grandchild then the amounts remaining in the savings account are:

However, starting with 8000 (= 20³) and giving (163, 278, 388, 67, 104, 112, 13, 14), leaves amounts of:

One way to rescue the puzzle is to exclude 1 as an amount given to the grandchildren (or remaining in the account).

Another way is to require that none of the amounts given to any grandchild is itself a perfect cube.

Either of these restrictions eliminate the first solution, leaving a unique answer of £ 388.

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## GeoffR 8:28 am

on10 May 2020 Permalink |LikeLike