December 2006
Shapes
A rectangular metal box with a base 2m by 3m and a height of 2m is filled with water to a height of 40cm. Five cubes, each 1m by 1m by 1m, are lowered into the box so that they all rest on the bottom. What is the height of the water in the box now?
Explain how you know that this is the right answer.
Numbers
The diagram shows a 5 by 5 grid with several greater than signs.
Place the numbers 1 to 5 in the grid so that each number appears once in each row and once in each column. In addition, the greater than signs must be true: for example, the number in the top left-hand corner must be greater that the number immediately below it.
Try to explain your reasoning, at least for the first few numbers.
Algebra
The diagram shows a magic square. The total of the three numbers in each row, each column, and each main diagonal should be the same.
Can you fill in the rest of the square?
(Hint: Either let the total be t and work out the other values in the square in terms of t, or let the numbers in the middle row be x, y, and 6, and work it out from there.)
Miscellaneous
The diagram below shows a 6 by 6 grid in which each cell has to be occupied by one of the four letters, A, B, C, and D.
No letter can be horizontally or vertically adjacent to itself.
The tables above and to the left of the grid indicate how many times each letter appears in that column or row.
Can you fill in the grid?
