## Teaser 2927: On a roll

**From The Sunday Times, 28th October 2018** [link]

My son saw the “average sheets per roll” printed on our new pack of four toilet rolls. Curious, he counted each roll’s total sheets by overlaying lines of ten sheets to and fro, tallying layers, and then adding any extra sheets. These totals were four consecutive three-figure numbers, including the printed “average sheets per roll”. For each total, he noticed that there was the same single-figure number of choices for a number of sheets per layer, requiring at least two layers, which would leave no extra sheets. Re-counting the toilet roll with the “average sheets per roll”, he used a two-figure number of sheets per layer and tallied a two-figure number of such layers, with no extras.

What was the “average sheets per roll”?

[teaser2927]

## Jim Randell 9:10 am

on3 April 2019 Permalink |This Python program runs in 81ms.

Run:[ @repl.it ]Solution:The average sheets per roll number is 242.The four toilet rolls have 242, 243, 244, 245 sheets. (So the average number of sheets per roll in the pack is higher that the stated average sheets per roll).

Each roll has 5 ways to lay out the sheets in piles:

And the “average” roll (with 242 sheets) can be laid out as 11× 22 sheets, or 22× 11 sheets.

This is the only solution with 3-digit values.

I checked on a 4-pack of loo rolls we have in the house, and they had an average sheet count of 221, which is not too far from the answer.

For 4-digit values for the number of sheets per roll there are 3 different sequences that work, but only one of these has a unique answer for the average sheet count.

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