## Teaser 2912: Be crafty and box clever

**From The Sunday Times, 15th July 2018** [link]

Tyson, a crafter, had several cylindrical enclosed boxes (with heights equal to diameters) in two sizes. Each box’s surface area (including top and bottom) was a three-figure integer in square centimetres, and the larger boxes had a radius which was a whole number multiple of that of the smaller boxes.

He kept them all in one vertical stack, to within one centimetre of the top of their metre-deep storage bin. Whilst working out fabric-cutting schemes, Tyson found that all the bigger boxes’ surface areas combined equalled all the smaller boxes’ surface areas combined and that the total surface area (in square centimetres) of all the boxes had all its numerals in ascending order.

What was the total surface area of all the boxes, in square centimetres?

[teaser2912]

## Jim Randell 12:19 pm

on22 May 2019 Permalink |The surface areas of the two different types of cylinders are both 3 figure integers. And the larger cylinder is a scaled up version of the smaller cylinder by a whole number. The smallest possible scaling factor is 2, which would make the surface area of the larger cylinder 4 times the surface area of the smaller cylinder. So the surface area of the smaller cylinder (

a) cannot be more than 249.This Python program starts by considering possible values for

aand then scaling up this value (by a factor ofk²) to giveAthe surface area of the larger cylinder.It then consider possible values for

N, the number of larger cylinders, and corresponding value forn, the number of smaller cylinders, until it finds suitable numbers that when stacked are between 99 and 100 cm high.It runs in 73ms.

Run:[ @repl.it ]Solution:The total surface area of the boxes is 3456 square centimetres.The smaller cylinders have a surface area of 144cm², with a corresponding diameter of about 5.53cm, and there are 12 of them, making a height of about 66.33cm.

The larger cylinders are scaled up by a factor of 2 (the scaling factor can only be 2 or 3), so have a surface area of 576cm², and a diameter of about 11.06cm. There are 3 of them, which stack to a height of 33.17cm.

The total stacked height of the 15 cylinders is about 99.50cm.

The total surface area of the 15 cylinders is 3456cm².

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