October 2004
Shapes
The diagram shows a hexomino, a shape formed from 6 squares.
There are 35 possible hexominoes. (Rotations and reflections of a shape are not counted as different.) How many of them could be used as the net of a cube (i.e. how many of them, if folded up, could form a cube)?
Numbers
In a (normal) magic square, each row, column, and diagonal has the same sum. For example, the diagram shows a 3 by 3 magic square, which contains 9 distinct positive integers.
Can you find a 3 by 3 multiplication magic square using 9 distinct positive integers where each row, column, and diagonal has the same product?
Extension
Try to find the magic square which has the smallest product.
Algebra
Sarah was out walking in the countryside when she came to a wooden bridge across a stream. As she approached it, out popped a troll. ‘Halt!’ he said, ‘I am the Double Crossing Troll. If you cross this bridge, I will double the money you have in your pocket. But then you must pay me my fee.’
‘How much?’ asked Sarah, and the troll told her. Sarah checked the money in her pocket. ‘OK’, she said, and marched bravely across the bridge. When she got to the other side, she felt in her pocket, and sure enough her money had doubled. She threw the troll his fee. She was about to go on her way, when a thought occurred to her. She walked back across the bridge, and checked her pocket. Yes, her money had doubled again. So she threw the troll his fee, and crossed for a third time. Again her money doubled, and after paying the troll again, she counted it up. ‘Right’, she thought, ‘I’ve now got exactly twice as much as I started with. So if I just keep crossing the bridge all afternoon …’
But then she remembered all the fairy tales she had read about what happened to people who were too greedy. ‘All right’, she thought, ‘I’ll just cross 3 more times, and then I’ll go home.’ And that is what she did. Each time, her money doubled, and then she paid the troll. When she got home, she counted her money, and found she had … how many times what she had set off with?
Miscellaneous
ABC School has 12 teachers, whose pictures are displayed in a frame on the wall. Given the following clues, can you work out whose picture goes where?
- Miss Abacus is to the right of Mr Boardrubber and above Mrs Cane.
- Mrs Dance is to the right of Miss Exercise and above Mr Felttip.
- Miss Games is below Mrs Cane, and to the left of Mrs Dance.
- Mrs Homework is above Miss Exercise and to the right of Mr Ink.
- Mr Boardrubber is below and to the right of Mrs Latin.
- Mr Felttip is to the right of Miss Abacus and above Miss Junior.
- Mrs Cane is to the right of Mrs Dance and above Miss Knowalot.
- Mr Ink is to the right of Miss Junior and below Mrs Latin.
(Note that ‘above’ does not necessarily mean directly above, and ‘to the right of’ does not necessarily mean directly to the right of. For example, picture 3 is above picture 9, and to the left of picture 12.)