Challenge Answers 2004
1. 990 teachers.
2. 3 parents and 2 each of the other categories.
3. 31 lines
4. 1260 m
5. 28 floors. The best way is to take the 10 from the 1st floor to the 4th, and the 10 from the 4th floor to the 7th; then go back down to pick up people from the 2nd and 5th, and finally return for those on the 3rd and 6th 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!