Problem 1008
After a track and field event, coach Larry Bambel arranges the young players bibs, numbered 1 to 2341 (included), into several crates, so that a number and his triple are never in the same box.- 1A. What is the largest possible number of bibs in the same box?
Larry takes the biggest number in the A, the smallest in the B, records their gap (in absolute value) and puts both bibs in a box; he starts again with the remaining bibs and adds the difference to the previous one, and so on until the piles are empty.
Following a similar process, Larry then ranks the under-14 bibs, less numerous than the under-16 bibs, numbered 1 to 2P. The total difference between the two categories (under-16 + under-14) is 2341.
- 2A and 2B. What is the number 2N of under-16 (2A) and the number 2P of under-14 (2B)?
For this problem, I made a program for question 2 only. Note that the statement suggests that the result obtained by summing the absolute values of the differences does not vary according to the distribution of bibs in boxes A and B. We can begin by checking this exhaustively in an increasing manner on the values of N, until the time becomes prohibitive (select_recurse() function). We note of course that this number is N2. Then, it is sufficient to test all the values of N and P that verify the statement.
The program can be found here.
The program can be found here.
- 1A: The largest possible number of bibs in a crate is 1756.
- 2A: There are 92 under-16 and 30 under-14.