Answer to August 23, 1999 Problem

The Handley Sports Problem

    Of the members of four boys' athletic teams at Handley,
    41 are on the football team,
    15 are on the basketball team,
    21 are on the baseball team,
    and 32 are on the wrestling team.

  • No wrestler is also a basketball player
  • 9 play football and baseball.
  • 5 wrestle and play baseball.
  • 12 wrestle and play football.
  • 6 play basketball and football.
  • 7 play basketball and baseball.
  • 3 play on all the teams except basketball.
  • 4 play on all the teams except wrestling.

How many individuals are involved in these four sports?

Solution to the Problem:

There are 77 athletes on the four Handley teams.

To solve this problem, a Venn diagram (or Euler circles) would be useful. Draw three circles which intersect each other (there should be 7 regions formed) and then draw one more circle which intersects only two of those circles (there should now be 11 regions). Each circle represents one of the sports. The 11 regions correspond to the 11 combinations of sports that could be played.

Begin by finding a region where three circles intersect. This could represent the number of athletes who play basketball, football, and baseball. Put the number 4 in that region.

Now find the other region where three circles intersect. This represents the athletes who are on the wrestling, baseball, and football teams. Put a 3 in that region.

Between those two regions is a region representing the players on just the football and baseball teams. Since there are 9 who play football and baseball, you should put 2 in this region (9 minus 4 minus 3) because 2 play only football and baseball.

Continue with this logic and you should get the following results:
4 are on the basketball, baseball, and football teams.
3 are on the wrestling, baseball, and football teams.
2 are on just the baseball and football teams.
2 are on just the wrestling and baseball teams.
9 are on just the wrestling and football teams.
2 are on just the basketball and football teams.
3 are on just the baseball and basketball teams.
7 are on just the baseball team.
6 are on just the basketball team.
21 are on just the football team.
18 are on just the wrestling team.

The sum of these 11 numbers is 77.


Correctly solved by:

1. Tom Marino Winchester, VA
2. Josh Grewal Winchester, VA
3. Jon Pence Winchester, VA
4. Richard Mocarski Winchester, VA