## Teaser 2926: Coins of Obscura

**From The Sunday Times, 21st October 2018** [link]

George and Martha were on holiday in Obscura where coins are circular with radii equal to their whole number value in scurats. They had coffees and George took out some loose change. He placed one 10 scurat and two 15 scurat coins on the table so that each coin touched the other two and a smaller coin between them that touched all three. He then placed a fifth coin on top of these four which exactly covered them and the total value of the five coins was exactly enough to pay the bill.

How much was the coffee bill?

[teaser2926]

## Jim Randell 8:02 am

on5 April 2019 Permalink |I think by “exactly covered” the setter means that the fifth coin was the smallest possible circle that would cover the original three coins.

If you’ve encountered “Soddy Circles” the fourth and fifth coins correspond to the inner and outer Soddy circles formed from the first three coins.

There is a (relatively) straightforward expression that can be used to calculate the radii of these circles (see: [ Descartes Theorem ]).

Here is a useful reference for Soddy Circles: [ link ]

This Python program runs in 89ms.

Run:[ @repl.it ]Solution:The bill came to 72 scurats.The additional coins have values of 2 and 30.

Here is a diagram. The original coins are shown in black, the inner and outer Soddy Circles (corresponding to the fourth and fifth coins) are shown in red:

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