Scheduling Teams of Persons

Dale Pleticha
Physics Department
Gordon College

Schedules in which any two persons are paired just once: Overview

In each of the schedules listed at the right, the total number of persons n is divided into teams, with m persons on each team. On each day of the schedule, each of the n persons is assigned to one team. In the course of the entire schedule, each person is teamed with each of the other (n−1) persons exactly one time. Schedules like these are called resolvable balanced incomplete block designs in design theory, which is a branch of combinatorial mathematics. Circular diagrams can help reveal the structure of these schedules.

The number of persons n divided by the number of persons per team m equals the number of teams. Furthermore, (n−1) divided by (m−1) equals the number of days needed to enable each person to work with every other person once. These schedules assume n/m and (n−1)/(m−1) are both integers. Consequently, for pairs of persons (m = 2), n must equal 2, 4, 6, 8, … . For triples (m = 3), n must equal 3, 9, 15, 21, … . For quadruples (m = 4), n must equal 4, 16, 28, 40 … . For quintuples (m = 5), n must equal 5, 25, 45, 65. And for sextuples (m = 6), n must equal 6, 36, 66, 96 … . However, even for some of these (m,n) values, schedules may not be possible. For example, a schedule for m = 6, n = 36 is not possible .

In the schedules listed at the right, m = 2, 3, 4, 5, and n is less than or equal to 40.

For the teams size you want, the schedules given here probably will not have exactly the number of persons and days that you have. You will usually have to choose a schedule which approximately fits your situation, and then, for example, adapt it by having one or more teams be slightly larger or smaller than the other teams, or by repeating some of the days of the schedule if it is too short.

How are schedules like these devised? Here are some references arranged from the introductory to the more technical:

   Martin Gardner, The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications, (Copernicus, New York, 1997) ch. 8.

   W. W. Rouse Ball, Mathematical Recreations and Essays, 11th edition, revised by H. S. M. Coxeter (Macmillan, New York, 1960), ch. X.

   Marshall Hall, Jr., Combinatorial Theory, 2nd edition (Wiley, New York 1986), ch. 15.


all in a Word document

22 persons
4 persons     24 persons
6 persons     26 persons
8 persons     28 persons
10 persons     30 persons
12 persons     32 persons
14 persons     34 persons
16 persons     36 persons
18 persons     38 persons
20 persons     40 persons

9 persons
15 persons
21 persons
27 persons
33 persons
39 persons

16 persons
28 persons
40 persons

25 persons