20-CS-110-001 Introduction to Computer Science Fall 2010

Solution to Track Meet Puzzle

Let a+b+c equal the number of 1st place points, a+b equal the number of 2nd place points, and a equal the number of 3rd place points. The total number of points per event is 3a+2b+c and the total number of points for the meet is 22+9+9 = 40 = n(3a+2b+c).

Because there is one competitor per school per event, each school must have a 1st, 2nd, or 3rd place finish in each event.

Possible values for n(point sum) are 10(4), 8(5), 5(8), 4(10)

Possible value for sums: >= 6 (3 2 1) hence n=5 or n=4.

Try n=5, sum=8, possible sums: 1+2+5 1+3+4
1+2+5:
 Can we get 9 points with one win? 1 win, 2 2nd place - no because we need 5 events 1 win, 4 3rd place - OK, number of awards is 5 4 2nd place, 1 3rd place - OK, number of awards is 5 Results for Washington: Lincoln must have a), Roosevelt b) Washington: 4 wins, 1 2nd, 0 3rd - OK, number of awards is 5 sum = 20+2 = 22
1+3+4:
 Can we get 9 points with two wins?    2 wins 1, 3rd - no since number of awards must be 5 Can we get 9 points with one win?    1 win, 1 2nd, 2 3rd - no, number of awards must be 5

Try n=4, sum = 10, possible sums = 1+2+7, 1+3+6, 1+4+5, 2+3+5
1+2+7:
 Can we get 9 points with 1 win? 1 win, 1 2nd - no, number of awards must be 4 1 win, 2 3rd - no, number of awards must be 4
1+3+6:
 Can we get 9 points with 1 win? 1 win, 1 2nd - no, number of awards must be 4 1 win, 3 3rd - OK Can we get 9 points with 0 wins? 3 2nd - no, number of awards must be 4 2 2nd, 3 3rd - no, number of awards must be 4 Cannot have L & R with 9 since #3rd is then 6
1+4+5:
 Can we get 9 points with 1 win? 1 win, 1 2nd - no, number of awards must be 4 1 win, 4 3rd - no, number of awards must be 4
2+3+5:
 Can we get 9 points with 1 win? 1 win, 2 3rd - no, number of awards must be 4

The only results possible are n=5, points=1,2,5
Washington: 4 wins, 1 2nd (22)
Lincoln: 1 win, 4 3rd (9)
Roosevelt: 4 2nd, 1 3rd (9)

Since Lincoln wins shot-put and nothing else, and Roosevelt wins nothing, only Washington could have won the long-jump.