December 2005
Shapes
Alice has seven strings of Christmas lights which she likes to arrange in her window. Each string has seven lights, all of different colours. Last year, she worked out how to arrange them in a 7 by 7 square pattern so that there was one light of each colour in each row and in each column.
She drew a diagram to remind herself how she did it, but unfortunately when she got it out this year it had become damp and faded, so that she could only make out where the strings had been placed, and a few of the positions of the colours.
Can you help Alice? Show how she can fill in this grid with one light of each colour in each row, each column, and each string (or polyomino) of seven squares.
Numbers
The diagram shows an addition sum. Each letter stands for a different digit. (So if ‘A’ represents 5, then ‘M’ cannot also represent 5.) A letter stands for the same digit everywhere it occurs. (So if the ‘M’ in ‘MERRY’ represents 5, then the ‘M’ in ‘XMAS’ also represents 5.)
Can you work out what digit each letter stands for?
Algebra
Three of Santa’s elves (Arwel, Bronwen, and Caedmon) were counting how many mince pies they had collected while helping Santa deliver his presents. They decided to even them out among them. First, Bronwen gave Arwel as many pies as he (Arwel) already had. Then, Caedmon gave Bronwen as many pies as she (Bronwen) had left after giving some to Arwel. Then, Arwel gave Caedmon as many pies as Caedmon had left after giving some to Bronwen. After all this, they discovered that they each had the same number of pies.
If they started with 240 pies between them, how many did each one start with?
Miscellaneous
The diagram below shows a chess board with a letter on each square. Find a Knight’s Tour which spells ‘Merry Christmas’.
(A Knight’s Tour is a series of moves which could be made by a Knight in chess.)