February 2005
Shapes
The diagram shows a 3 by 3 square with a unit square attached to one of the corners.
Show how to cut this shape into three pieces which can be fitted together to make a perfect square.
Numbers
Alice and Betty went to the corner shop and bought two boxes of sweets to give to their boyfriends on Valentine’s Day.
‘£4.90, please’, said the shopkeeper, and Alice paid the money.
Outside the shop, Betty said, ‘Didn’t you notice? He multiplied the amounts on his calculator instead of adding them.’
‘It doesn’t matter’, said Alice. ‘We paid the right amount anyway.’
What were the prices of the two boxes of sweets?
Algebra
Alice has been collecting her small change (1p, 2p, and 5p coins) to give to charity. When she counted it up, she found that she had exactly 100 coins. There were twice as many 1p coins as 5p coins, and the total amount was £2.24.
How many of each type of coin did she have?
Miscellaneous
The diagram shows a 5 by 5 grid.
Show how to place ten counters on the squares of this grid so that no three counters lie in a straight line, horizontally, vertically, or at any angle.
Extension
There are five possible solutions, not counting rotations and reflections as different. How many of them can you find?