**Challenge
Answers 2004**

Thank
you for your interest in Challenge 2004. You can find the questions here.
Please email me at p.m.strickland@livjm.ac.uk if you have
any comments or questions,

Best wishes,

Paul Strickland.

1.
990 teachers.

2. 3 parents and 2 each of the other
categories.

10 ways

3. 31 lines

4. 1260 m

5. 28 floors. The best way is to take
the 10 from the 1^{st} floor to the 4^{th}, and the 10 from the
4^{th} floor to the 7^{th}; then go back down to pick up people
from the 2^{nd} and 5^{th}, and finally return for those on the
3^{rd} and 6^{th} floor.

6. With six lines you can make 36 angles
of 90° or 24 of 60°.

With twelve you can make 144 angles of 90° or 96 of 60°. See the picture below for the 96 angles;

7. Just looking at the numbers;

37 = 10 + 9 ´ 3 = 10
+ 6 + 7 ´ 3 = 10 + 2 ´ 6 + 5 ´ 3.

= 10 + 3 ´ 6 + 3 ´ 3 = 10 + 4 ´ 6 + 3.

You can make a 37-hexagon out of a
10-triangle and 9 3-triangles, either by putting the 10-triangle (marked by Ts)
at the edge of the hexagon or in the centre;

T T T T . . . .

. T T T . . . T . .

. . T T . . . . T T . .

. . . T . . . or . . T T T . .

. . . . . . . T T T T .

. . . . . . . . . .

. . . . . . . .

In each case (and in all the others in
fact!), there is only one way to fill in the remaining 3-triangles.

Surprisingly there is no way of completing the
hexagon using an odd number of 6-triangles. Perhaps when we have finished
marking the scripts we will find someone who has got a good reason why ? watch
this space! But with two or four 6-triangles we can do it. We need the
10-triangle along an edge; then the 3-triangles (marked with S) can go along
any of the other edges, as in these two examples;

T T T T T T T T

S T T
T . . T T T S

S S T T
. . . . T T S S

S S S T
. . . or S S S T S S S

. . S . . . S S S S S S

. S S . . S S S S S

S S S . S S S S

Again, the 3-triangles can only fit in in
one way each time. So how many ways are there? If you ignore solutions which
are reflections or rotations of each other, there are 2 + 6 + 3 = 11 ways of
solving the puzzle. If you found two or three, then very well done!