There are many traditional puzzles and problems associated with the chessboard. One of the best known is the ‘Knights Tour’. What paths can a knight follow on a chessboard, using the traditional knight’s move, so that it visits each square once only? Two interesting solutions are given below. The first was found by de Moivre using a strategy of moving in one direction and keeping as close to the boundary as possible:

(a) pattern shown by numbers

(b) pattern shown by lines

The second, found by Euler, forms an 8 x 8 magic square (see below). This diagram could be copied onto squared paper and the moves shown as straight lines as in de Moivre’s diagram.

There are 4860 ways of placing five queens on a chessboard so that all squares are either occupied or commanded. This problem can be extended to consideration of the smallest number of queens needed to command an n x n chessboard.

A 4 x 4 board needs 2 queens.
A 5 x 5 board needs 3 queens.
A 6 x 6 board needs 3 queens.
A 7 x 7 board needs 4 queens.
An 8 x 8 board needs 5 queens.

Further discussion of chess problems can be found at:
http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Mathematical_games.html#44

Palindromic Words and Sentences

Here are some word palindromes:

ANNA
BOB
DAD
DEED
DID
MUM
NOON
OXO
POP
TOOT

There are a number of well-known sentence palindromes, although these are difficult to create. Here are some favourites:

When Adam first met Eve in the Garden of Eden he might have said: ‘MADAM, I’M ADAM’.
A French engineer called Ferdinand de Lesseps had the idea of digging through Panama to provide passage for ships. He did not quite finish the job but his epitaph might have been: ‘A MAN, A PLAN, A CANAL: PANAMA!’
‘I’M A LASAGNA!’ SANG A SALAMI.
The marketing industry would surely love: ‘A TOYOTA. RACE FAST, SAFE CAR. A TOYOTA.’

There are some more examples at http://www.jps.net/msyu/palindromes/. Reading University’s list of what they claim may be the world’s first palindromic URLs has more information and good links to other sites. The address is:
http://www.rdg.ac.uk/~sssbownj/jnwobsss~/ku.ca.gdr.www//:ptth/.

The Guard’s Problem

The Guard’s problem is traditional and is based on various legends concerning the invention of chess. It is said that a Persian shah (or an Indian maharajah, or a Chinese emperor...) was so impressed with the game that he offered a reward to the inventor. The inventor asked only for: a grain of wheat (or rice) on the first square, 2 grains on the next square, 4 grains on the next square, and so on. Is this more or less than the amount of matter in the Solar System? The Galaxy? The Universe? What if the request was not for grains, but for bags? Catering sacks? Lorryloads? A modern take on the same problem is given at http://math.rice.edu/~lanius/pro/rich.html. There are also good links from here to other related material and notes for teachers.