Recreational Maths


About

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, ..., n2 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!

Magic Square

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.

Pan1
Pan2
Example1
Example2
Example3
Example4
Example5
Example6

Try the below Pandiagonal Magic Sqaure. Remember each number 1-16 can only appear once!

Pandiagonal Magic Square

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!

Queens
1/12 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).

alphabet

Correspondents

Geoffrey Thomas Bennett

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

   

Cell-Aggregates

   

All Correspondence Links

Claxton Fidler

   

Eric Harold Neville

   

John Marshall

   

Alfred North Whitehead

   

Charles Robert Darling

   

Peter Guthrie Tait

   

William Peddie

   

Geoffrey Thomas Bennett

   

Dorothy Wrinch

   


Main Index Biographies Index


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Cammy Sriram and Edward Smith © July 2019
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School of Mathematics and Statistics
University of St Andrews, Scotland

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http://www.mcs.st-and.ac.uk/~dat/