May 2006
Shapes
The diagram shows a grid filled with polyominoes. (A polyomino is a shape formed by one or more squares.) Each square in a polyomino contains an integer, which represents the number of squares in that polyomino. The polyominoes with the same number of squares do not touch each other, except at a corner.
Can you complete the 9 by 9 grid below (by filling each square with a number and marking the edges of each polyomino) according to the rules above?
Numbers
Here are some musical sums. Each letter represents a different digit from 0 to 9. Each letter represents the same digit every time it occurs. Can you replace the letters with the correct digits?
RE + MI = FA
DO + SI = MI
LA + SI = SOL
Algebra
Little Jane is playing with the balancing scales from her kitchen. She is putting the plastic scales from her toy zoo onto the scales to see what will balance. This is what she finds:
- One elephant and one giraffe will balance three zebras.
- One giraffe and one zebra will balance four antelopes.
- One antelope and two zebras will balance one elephant.
How many antelopes would it take to balance an elephant?
Miscellaneous
This is a ‘skyscraper’ puzzle. Each of the squares in the 5 by 5 grid is occupied by a building between 1 and 5 stories tall. Each building in a row or column is a different height; therefore each height occurs only once in each row and in each column.
The numbers around the edge of the box indicate how many buildings can be seen in that row or column, when looking from that side. For example, if the buildings in a row were in the order 2 5 3 4 1, then the number at the left of the row would be 2, since only the 2-storey and 5-storey buildings could be seen, the other three buildings being hidden by the 5-storey building. The number at the right of the row would be 3, because the 1-storey, 4-storey, and 5-storey buildings would be visible; the 3-storey building would be hidden by the 4-story building and the 2-storey building would be hidden by several of the buildings in front of it.
Can you fill in the sizes of the buildings in the puzzle?