Nobody solved this so it becomes another Tough Nut. Dorothy Winn
of Madras College, St Andrew's and Maren Farra & Corinna Calori
of Camden School for Girls used the same notation denoting the
people by the numbers 1, 2, 3,...? in ascending order of height. As
each person at the back must be taller than the person directly in
front of them and, along the rows, the heights must increase from
left to right the 1 must be in front on the left and the tallest
must be at the back on the right. Dorothy gave the results for 2,
4, 6, and 8 people in the following table.
| Number of people |
Number of arrangements |
Diagrams |
| 2 |
1 |
|
| 4 |
2 |
|
| 6 |
5 |
2 4 6
1 3 5 |
3 4 6
1 2 5 |
2 5 6
1 3 4 |
3 5 6
1 2 4 |
4 5 6
1 2 3 |
|
| 8 |
14 |
2 4 6 8
1 3 5 7 |
3 4 6 8
1 2 5 7 |
2 5 6 8
1 3 4 7 |
3 5 6 8
1 2 4 7 |
4 5 6 8
1 2 3 7 |
2 4 7 8
1 3 5 6 |
3 4 7 8
1 2 5 6 |
2 5 7 8
1 3 4 6 |
3 5 7 8
1 2 4 6 |
4 5 7 8
1 2 3 6 |
2 6 7 8
1 3 4 5 |
3 6 7 8
1 2 4 5 |
4 6 7 8
1 2 3 5 |
5 6 7 8
1 2 3 4 |
|
Many people think that because the sequence 1, 2, 5, 14 ? goes
up in powers of 3 (with differences 1, 3 and 9) the next difference
will be 3 cubed to give the next number 14 + 27 = 41. Maths is full
of patterns but, when you think you spot a pattern you have to
prove it always works. If you count the arrangements for 10 people
the answer is 42 and not 41. Can you find them all?
More importantly can you explain why there are 42
arrangements for two teams of 5 players and find a formula for the
general case?
Peter Conlon, age 15, The Worth School, Sussex has spotted that
these are the first few Catalan numbers. Well spotted Peter, and
you are right. We still lack a proof that the sequence of numbers
for more people in the photograph is, in general, the sequence of
Catalan numbers. Checking a few cases is not suifficient to prove
that a statement is ALWAYS true.