“Lightly grease a 20x20cm baking tin with butter and spoon in the mixture. Press into the corners with the back of a spoon so the mixture is flat and score into 12 squares.”
- BBC Good Food flapjack recipe by user nicolajlittle
Hang on a minute - I thought, mid-baking. It doesn’t say what size squares – and they clearly can’t all be the same size. (The pedestrian solution of ‘perhaps it means rectangles’ didn’t appeal.)
I had inadvertently stumbled across a classic Henry Dudeney puzzle. And, uniquely for the Dictionary of Mathematical Eponymy, it’s named after a fictitious character.
What is Mrs Perkins’ Quilt?
““For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches.”
- Henry Dudeney, Amusements in Mathematics problem 173.
Although it’s Dudeney’s puzzle that gives the quilt its name, it’s a few years older than the 1917 puzzle book it appears in - Singmaster says it was published by Sam Loyd in 1907.
The answer given – you may wish to find it yourself, for a small challenge, or prove that it’s optimal, for a bigger one – involves eleven smaller squares arranged in a particular way. The term ‘Mrs Perkins’ Quilt’ has come to mean any dissection of a square into smaller squares with integer sides.
There’s also a link to the Scottish Book: problem 59, due to Ruziewicz, asks “Can a square be divided into squares that all have different dimensions?” - adding an extra constraint to the question.
Why is it important?
As with several recent entries, I’m not certain that Mrs Perkins’ Quilt is, in itself, especially important. However, it’s interesting. It’s not a typical geometry problem, or a typical combinatorics problem, or a typical algebra problem; it requires a different way of thinking, new notations, new ways of writing programs. And it’s involved some Seriously Big Names.
For example, one of the earliest approaches to Problem 59 involved turning the square into an equivalent electrical circuit and using Kirkhoff’s laws and circuit decomposition techniques - the pictures of the resulting solutions were designated as Smith diagrams ((I’m a huge fan of insightful diagrams.)) despite Smith’s protests - by Bill Tutte ((Yes, that Bill Tutte)), in an article popularised by Martin Gardner ((Yes, there’s only one Martin Gardner)).
It’s worth noting that the Mrs Perkins’ Quilt problem for general square-size $n$ is unsolved - there isn’t a known method for determining the number of squares in the optimal solution, or how they’re laid out. It is known that the number of squares required is of order $\log(n)$ - this was proved by Conway ((Yes, that Conway)) and Trustum.
Who was Potipher Perkins?
She was made up for the purposes of a puzzle. Good name, though. She sounds like someone who would appreciate a flapjack.
- See Alaric Stephen’s article here, and Ed Pegg Jr’s summary of the state of play here. There’s a detailed write-up of the method in Smith diagrams here.
A selection of other posts
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