During the correspondence of D’Arcy and Geoffrey Thomas Bennett, the two discussed many recreational mathematical puzzles. As D’Arcy was, in his own words, “ignorant” of mathematics, it’s interesting that he took such an interest in these problems. Perhaps he knew more than he let show!
Here are a few examples of the problems discussed, see if you can solve them!
Magic Squares
The first "fun" problem discussed was that of Magic Squares (ms26217), which are solved as follows: Fill an nxn square grid with numbers in the range 1, 2, ..., n^{2} so that each cell contains a different number and the sum of the numbers in each row, column and diagonal is equal (the number they equal is called the magic constant).
Try the set of 3x3 squares below!
Upon hearing that D'Arcy had been playing with the Squares, Bennett suggested that D'Arcy try the more mathematical (and more difficult) Pandiagonal Magic squares (ms26218) in which the broken diagonals also need to add up to the magic constant. See Bennett's explanation of this problem from ms26059 and the broken diagonals of a 4x4 square below.
Try the below Pandiagonal Magic Sqaure. Remember each number 116 can only appear once!
After a fair bit of explanation, D'Arcy managed to get to grips with the Pandiagonals (ms26220).
Eight Queens Problem
D'Arcy brought this problem up with Bennett in early 1941 (ms26247) while in bed recovering from bronchitis. The two discussed the problem and Bennett told D'Arcy of how in his youth he had sent a proof for the impossibility of there being only 8 solutions in a letter to the Messenger but that credit was given to someone else later (ms26135).
The problem is as follows: place eight chess queens on an 8×8 chessboard so that no two queens threaten each other. There are 12 solutions, how many can you get?
Hint: Here’s 1 of the 12 solutions!
Sequences
Bennett and D’Arcy also discussed sequences, such as the the Fibonacci Sequence and one which Bennett devised for the position of the vowels in the alphabet, which is shown below (ms26165).
Introduction

Overview & D'Arcy's Life

On Growth and Form

Heilmann & Shufeldt

Maths of Transformations

Correspondence

D'Arcy and Mathematics

Coordinate Transformations

Logarithmic Spirals

Forms of Cells

Forms and Mechanical efficiency

Shrinkage

Wartime and D'Arcy

The Leg as a Pendulum

Recreational Maths

Fibonacci Sequence

CellAggregates

Claxton Fidler

Eric Harold Neville

John Marshall

Alfred North Whitehead

Charles Robert Darling

Peter Guthrie Tait

William Peddie

Geoffrey Thomas Bennett

Dorothy Wrinch



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Cammy Sriram and Edward Smith © July 2019 Except where otherwise indicated, the text in this work by Cammy Sriram and Edd Smith is licensed under a Creative Commons AttributionShareAlike 4.0 International License.  School of Mathematics and Statistics
University of St Andrews, Scotland 
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