September 2004
Shapes
The diagram shows five pentominoes arranged in a square.
Can you place the numbers 1 to 5 in the squares of each pentomino, so that each number appears once in each pentomino, once in each row of the square, and once in each column of the square?
Explain the reasoning that you use.
Numbers
Place the numbers 1 to 8 in the unshaded squares of this grid so that on each of the four sides the middle number is the sum of the two corner numbers.
Explain the reasoning that you use.
Algebra
Alice was on her way to her grandmother’s house when she came to a wooden bridge across a stream. As she approached it, out popped a troll.
‘Halt’, he said, ‘To cross this bridge you must pay my fee.’
‘How much?’ asked Alice.
‘That depends’, said the troll. ‘The first two people to cross the bridge each morning just have to pay me all the money in their pocket. After that, each person has to pay as much as the two previous people added together.’
‘But that could be lots and lots’, cried Alice, ‘I might not have that much. How many people have already crossed today?’
‘You will be the sixth, if you cross’, said the troll, smiling to himself. ‘And the first person only had £2.’
Alice thought for a while. ‘It’s no use. I still can’t work it out’, she said.
‘All right’, said the Troll, ‘I’ll tell you. You’ll have to pay £17.’
Alice looked in her purse, and found that she had just enough. ‘But I’ll get here at daybreak next time, with two pence in my pocket!’ she thought.
After Alice pays him, how much money will the troll have taken today?
Explain how you work this out.
Miscellaneous
The diagram on the right shows a continuous line which passes through every square in the grid, except the squares which have been shaded in.
Can you draw a continuous line which passes through every square in the grid on the left, except the squares which have been shaded in?
Try to explain the main steps in the logic of your solution, even if you cannot explain every detail.