Answer to April 28, 2003 Problem

The Hymnal Problem

A hymn book contains 700 hymns. A bulletin board at the front of the church has four grooved rows on which the numbers of the hymns to be sung are placed.

What is the smallest number of plates, each carrying one figure, which must be kept in stock so that the numbers of any 4 hymns may be displayed in the four rows?
You may assume that the church never sings the same hymn twice on the same Sunday.

 

Solution to Problem:

Originally, I believed that the answer was 86 plates and the chart below shows how many of each number would be needed:

Plate Number How Many
0 8
1 9
2 9
3 9
4 9
5 9
6 9
7 8
8 8
9 8

However, John Funk sent in a very clever solution that requires the church to order only 81 plates! He had the Church secretary order 12 sixes and no nines (since you could invert the sixes to make nines), and 12 plates cover every possibility of four songs with all sixes or nines. So, I believe that there was only one correct solution this week, but I listed all the rest who thought like I did. Here is the chart showing how many of each number would be needed under John's plan:

Plate Number How Many
0 8
1 9
2 9
3 9
4 9
5 9
6 12
7 8
8 8
9 0

Correctly solved by:

1. Andrea Eberhard Columbus, Ohio
2. John Funk Ventura, California
3. William Funk San Antonio, Texas
4. Rick Jones Kennett Square, Pennsylvania
5. Michael Rodriguez Great Falls, Montana
6. Jeff Gaither Winchester, Virginia
7. Matt Stillwagon Winchester, Virginia
8. James Alarie University of Michigan -- Flint
Flint, Michigan
9. Misty Carlisle Winchester, Virginia
10. Richard Johnson La Jolla, California
11. Ben Reames Columbus, Georgia