Henry Ernest Dudeney came from a family which had a mathematical tradition and also a tradition of school teaching. Henry's father was a schoolmaster and his father, Henry's paternal grandfather, although he began life as a shepherd, taught himself mathematics and astronomy and left his life on the hills to become a schoolmaster in Lewes. Henry learnt to play chess at a young age and soon became interested in chess problems. From the age of nine he was composing problems and puzzles which he published in a local paper. Although he only had a basic education, never attending college, he had a particular interest in mathematics and studied mathematics and its history in his spare time. As he pointed out:
Date of death:
Place of death:
10 April 1857
Mayfield, Sussex, England
24 April 1930
Lewes, Sussex, England
The history of mathematical puzzles entails nothing short of the actual story of the beginnings and development of exact thinking in man.
Certainly, as he realised in reading about the history of mathematics, its development is closely linked with puzzle solving.
Dudeney worked as a clerk in the Civil Service from the age of 13 but continued to study mathematics and chess. He began to write articles for magazines and joined a group of authors which included Arthur Conan Doyle. At this stage he was doing well publishing mathematical puzzles under the pseudonym 'Sphinx'. In 1884 Dudeney married and his wife, a popular novelist of the day, helped to make the family very well off financially.
Sam Loyd started sending his puzzles to England in 1893 and a correspondence started between him and Dudeney. The two were the main creators of mathematical puzzles and recreations of their day and it was natural that they should exchange ideas. Of the two puzzle experts it was Dudeney who showed the more subtle mathematical skills. He sent a large number of his puzzles to Loyd and became very upset when Loyd began to publish them under his own name.
In Newing describes how one of Dudeney's daughters:
... recalled her father raging and seething with anger to such an extent that she was very frightened and, thereafter, equated Sam Loyd with the devil.
We have indicated that Dudeney had a mathematical talent and this is very clear looking at some of his famous puzzles. For example, one of the most famous of his geometrical puzzles is the 'haberdasher's problem' which asks how one can cut an equilateral triangle into four pieces that can be reassembled to form a square. He had a model made which was hinged in such a way that it could be formed into a square or an equilateral triangle. The Royal Society was interested in this geometrical novelty and in 1905 Dudeney demonstrated his geometrical puzzle at a meeting of the Society.
Dudeney contributed to the Strand Magazine for over 30 years, beginning after his collaboration with Loyd ended, and from around the same time he began publishing in Blighty, Cassell's, The Queen, Tit-Bits, and the Weekly Dispatch. Dudeney's very popular collections of mathematical puzzles The Canterbury Puzzles (1907), Amusements in Mathematics (1917), and Modern Puzzles published in 1926, contain a wealth of fascinating examples which would provide any teacher of mathematics with a treasure trove of material.
Since we are looking at the more mathematical aspects of Dudeney's work, the chapter headings of Amusements in Mathematics will prove interesting. These are: Arithmetical and Algebraical Problems; Geometrical Problems; Points and Lines Problems; Moving Counter Problems; Unicursal and Route Problems; Combination and Group Problems; Chessboard Problems; Measuring, Weighing, and Packing Puzzles; Crossing River Problems; Problems concerning Games; Puzzle Games; Magic Square Problems; The Paradox Party; Unclassified Problems.
He wrote about the psychology of puzzles in the Prefaces to some of his books:
The fact is that our lives are largely spent in solving puzzles; for what is a puzzle but a perplexing question? And from our childhood upwards we are perpetually asking questions or trying to answer them.
Again he wrote:
The solving of puzzles consists merely in the employment of our reasoning faculties, and our mental hospitals are built expressly for those unfortunate people who cannot solve puzzles.
One would have to say that there is rather a lot of evidence against this latter point of view!
You can read the Preface to Amusements in Mathematics
As we mentioned, chess was one of his first encounters with puzzles and it remained an interest throughout his life. He was a founding member of the British Chess Problem Society in 1918, chairing its first meeting. Like Loyd , Dudeney produced many non-standard chess problems such as one where the White pieces are in their initial position, while Black only has a King which is on its own initial square. The problem is to find a mate in 6 for White.
After Dudeney's death his wife helped edit a collection of his puzzles Puzzles and Curious Problems (1931) and later on she again helped edit a second collection which was entitled A Puzzle Mine.
References and show that Dudeney's puzzles are still of interest to many mathematicians. In a generalisation of Problem 229 in Martin Gardner's H E Dudeney's 536 puzzles and curious problems (1967) is discussed. In the following problem of Dudeney, posed first in 1905 and appearing in Amusements in mathematics (1917), is discussed:
Is it possible to seat n people at a round table on (n - 1)(n - 2)/2 occasions so that each person has the same pair of neighbours exactly once.
A proof is given in for n even, but the case of n odd still appears to be open.
Let us now look at a few more examples of Dudeney's puzzles. First one called Catch the hogs which is from The Canterbury Puzzles (1907):
In the illustration Hendrick (H) and Katrun (K) are seen engaged in the exhilarating sport of attempting the capture of a couple of hogs (BP the black pig, and WP the white pig). Why did they fail?
He then goes on to explain the rules of the game. Player one moves first and moves both Hendrick (H) and Katrun (K) one square each in any direction, but not diagonally. Player two then moves the hogs (BP and WP) each one square, again not diagonally. Try the game and see if Hendrick and Katrun can catch the hogs!
Player one will fail if K tries to capture WP and H tries to capture BP.
However, if K tries to capture BP and H tries to capture WP they will succeed.
It is a parity problem.')">HERE for the solution.
Here is another problem from The Canterbury Puzzles which is easily solved with a little mathematics:
It used to be told at St Edmondsbury that many years ago they were overrun with mice that the good abbot gave orders that all the cats from the country round should be obtained to exterminate the vermin. A record was kept, and at the end of the year it was found that every cat had killed an equal number of mice, and the total was exactly 1111111 mice. How many cats do you suppose there were?
Other puzzles simply reduced to systems of linear equations if a mathematical solution was sought. For example Problem 3 from Amusements in Mathematics:
Three countrymen met at a cattle market. "Look here, " said Hodge to Jakes, "I'll give you six of my pigs for one of your horses, and then you'll have twice as many animals here as I've got." "If that's your way of doing business," said Durrant to Hodge, "I'll give you fourteen of my sheep for a horse, and then you'll have three times as many animals as I." "Well, I'll go better than that, " said Jakes to Durrant; "I'll give you four cows for a horse, and then you'll have six times as many animals as I've got here."
No doubt this was a very primitive way of bartering animals, but it is an interesting little puzzle to discover just how many animals Jakes, Hodge and Durrant must have taken to the cattle market.
Problem 11 from the same book reduces to a quadratic equation:
'Twas last Bank Holiday, so I've been told,
Some cyclists rode abroad in glorious weather.
Resting at noon within a tavern old,
They all agreed to have a feast together.
"Put it all in one bill, mine host," they said,
"For every man an equal share will pay."
The bill was promptly on the table laid,
And four pounds was the reckoning that day.
But, sad to state, when they prepared to square,
'Twas found that two had sneaked outside and fled.
So, for two shillings more than his due share
Each honest man who had remained was bled.
They settled later with those rogues, no doubt.
How many were they when they first set out?
Dudeney invented something he called Verbal Arithmetic. One of his examples of this is the famous
S E N D
M O R E
M O N E Y
In this addition sum each letter represents a digit, different letters being different digits. Find the sum.
Finally let us give an example of a geometrical problem from Amusements in Mathematics.
This is the Joiner's problem:
A joiner had two pieces of wood of the shapes and relative proportions shown in the diagram. He wished to cut them into as few pieces as possible so that they could be fitted together, without waste, to form a perfectly square table-top. How could he have done it? There is no necessity to give measurements, for if the smaller piece (which is half a square) be made a little too large or small, it will not effect the method of solution.
The problem can be solved with only two cuts, creating five pieces as shown.')">HERE for the solution.
Dudeney's final illness is described in :
He had been in failing health for some time, but, even through his illness, he continued to write his articles for London journals.
Source:School of Mathematics and Statistics University of St Andrews, Scotland