January 2006
Shapes
The diagram shows three congruent pentominoes which have been placed together to form a shape with a line of symmetry.
Show how the five congruent pentominoes below can be placed together to make a different shape with a line of symmetry.
Give some indication of your reasoning.
Numbers
Welcome to 2006!
Can you insert the mathematical symbols + - × ÷ ( ) ! between the numbers below so that the resulting expression is equal to 2006?
8 7 6 5 4 3 2 1
Each symbol may be used as often as is required. The digits must remain in the order given.
(Hint: ‘3!’ is pronounced ‘3 factorial’ and means 3 × 2 × 1 = 6. Similarly, ‘4!’ means 4 × 3 × 2 × 1 = 24, and so on.)
AlgebraHere is a number trick: Think of a number with three digits, all different. Now make all the two-digit numbers that you can from the three digits of your number. (There should be six such two-digit numbers.) Add the six, two-digit numbers together. Now divide your answer by the sum of the digits of your original number. The number that you get is…?
Prove that you will always get the same number, no matter what three-digit number that you start with.
(Hint: In algebra, the two-digit number formed from the digits x and y is written ‘10x + y’, not ‘xy’, which of course means x × y.)
Miscellaneous
Five contestants were competing on a TV show. In the final round they were shown five ‘mystery’ photos and given five names. They had to match the photos with the names. The contestants wrote down their answers:
| Contestant | Photo 1 | Photo 2 | Photo 3 | Photo 4 | Photo 5 |
|---|---|---|---|---|---|
| Annie Adams | Andrew Flintoff | Tony Blair | Robbie Williams | Bob Geldof | Richard Branson |
| Betty Bean | Bob Geldof | Richard Branson | Andrew Flintoff | Robbie Williams | Tony Blair |
| Clare Clarke | Tony Blair | Bob Geldof | Robbie Williams | Richard Branson | Andrew Flintoff |
| Dionne Davis | Bob Geldof | Tony Blair | Andrew Flintoff | Robbie Williams | Richard Branson |
| Ellie Evans | Bob Geldof | Richard Branson | Robbie Williams | Andrew Flintoff | Tony Blair |
When the answers were given, it was found that no two contestants had the same number of correct answers. Which contestant went home with the first prize?