Neurobics

Latin Squares

Russell Dear

In the late 18th century the Swiss mathematician Leonard Euler (pronounced "oiler") systematically developed the theory of, what are today called, Latin squares. A Latin square is a square array of entities, usually letters or numbers, none of which occur more than once in each row or column. They are used in experimental design. Here, for example, are a couple of three-by-three Latin squares and a four-by-four one.

A

B

C

D

A

B

C

A

C

B

B

A

D

C

B

C

A

B

A

C

C

D

A

B

C

A

B

C

B

A

D

C

B

A

How many different three-by-three Latin squares are there, I wonder?

Two problems related to Latin squares might interest you.

The first is to see if you can arrange the nine letters (three of each of A, B and C) in a square so that there are two of the same letters in each row and column?

The second was a popular card puzzle in the 18th century. The idea is to see if you can arrange the sixteen highest cards of a standard pack (the aces, kings, queens and jacks) in a four-by-four square so that every column and row will contain all four values and all four suits. When you've done that, see if you can find a solution that has its two diagonals also showing one card of each value and one of each suit.

Henry Dudeney

Apart from crossword puzzles, only a few magazines and newspapers today feature mind-engaging problems such as logic puzzles and conundrums. In Victorian times however virtually all magazines included puzzles or problems for readers' entertainment. The selection was wide; from problems associated with parlour games such as go-moku, bridge and chess, to quite demanding brain teasers.

Many of the compilers of these puzzles were household names. One such, and probably the master of them all, was Henry Dudeney. He was born in England in 1847 and died there in 1930. He contributed puzzles to many English magazines such as The Strand, Tit-bits and the Weekly Dispatch. He also wrote three books, two of which, The Canterbury Puzzles and Amusements In Mathematics, are still available from Dover Publications. The Canterbury Puzzles was written in the style of Chaucer's Canterbury Tales with a group of travellers exchanging problems to pass the time on a journey.

There is no doubt that Dudeney had a good understanding of number theory on which many of his problems were based. He also had a flair for writing puzzles which were completely original and masterpieces of ingenuity. I have chosen two for you below.

Dudeney always titled his problems and set them in an appropriate context. The first problem from his Amusements In Mathematics is of historic interest as well as an exercise in arithmetic. Aren't you glad we've gone metric? [For those who don't remember pre-decimal currency, there are 12 pence to a shilling, five shillings to a crown.]

A Post-Office Perplexity

In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. I quite pitied a young lady in a branch post-office when a gentleman entered and deposited a crown on the counter with this request. "Please give me some twopenny stamps, six times as many penny stamps, and make up the rest of the money in twopence-halfpenny stamps."

For a moment she seemed bewildered, then her brain cleared, and with a smile she handed over stamps in exact fulfilment of the order. How long would it have taken you to think it out?

The next problem is on a related theme. It led to a more recent series of challenges in Scientific American on general problems related to folding sheets of stamps.

The Four Postage Stamps

You have just bought a three by four sheet of stamps and want to give a friend four, all joined together -- no stamp hanging on by a mere corner. In how many different ways is it possible to tear off the four stamps? You could give her 1, 2, 3, 4 or 1, 2, 6, 5 or 3, 7, 8, 12. See how many ways you can find altogether.

Answers

(1) 5 twopenny stamps, 30 penny stamps, and 8 twopence-halfpenny stamps

(2) There are 65 ways the four stamps may be torn from the sheet.

					Q	K	J	A
A	A	B			J	A	Q	K
B	C	B			A	J	Q	K
A	C	C			Q	K	A	J

Russell Dear is a Mathematician living in Invercargill