Almost 20 years ago, in the warm summer of 1996, I was earning my money as a developer at a well-known software company east of Munich when a colleague excitedly reported a simple-sounding, but apparently difficult-to-solve, problem.

The principal of a school was confronted with the task of dividing nine teachers, 27 female students, and 18 male students into nine groups that would change in such a way that the students would sit in class with different teachers and different classmates each day. Each group was to consist of exactly one teacher, two boys, and three girls, and nine group classes were to take place in different classrooms each day, with each class taught by one of the nine teachers. For how many days could the school go ahead with lessons without the group compositions overlapping? Maybe five?

"Nothing easier than that!" my colleagues and I shouted immediately. We put our project on hold for a while and started to type the first programs on our workstations. Figure 1 shows the first attempt with a simple algorithm that populates nine groups with the same teachers, and not-yet allocated students, every day. It keeps a record of who has already been together with whom and simply takes the next student from the not-allocated pool if there is an overlap.

Figure 1: If you populate the groups sequentially with students, you will not even manage two days without overlaps.

The teachers are enumerated with a letter i (for instructor) from 1 to 9, the boys from b1 to b18 (boy) and the girls with g1 to g27 (girl). For example, teacher i1 conducts the class in group room 1 on the first day with male students b1 and b2 and female students g1, g2, and g3:

day1 group1: i1 b1 b2 g1 g2 g3

On the second day, the algorithm in group 1 detects that boys b1 and b2 were already paired with teacher i1 the previous day and therefore assigns b3 and b5 to the latter (it can't use b4 because b4 attended class with b3 on the day before). And, it works this way for a while.

However, by the end of the second school day, it hits a brick wall, because there are no more female students available for group 7: g19, g20, and g21 were already with i7 in group 7 on the first day. Also, g22, g23, and g24 were already with b15 in group 8. And, g25, g26, and g27 were with b17 in group 9, so the strategy failed miserably!

Seriously Miscalculated

After several failed attempts, it slowly began to dawn on us that this probably required more brain and computing power than first imagined. So, we started to mistreat our brand new HPUX workstations with backtracking algorithms.

To everyone's surprise, however, we did not see any success after a few hours of merciless brute-force attacks, but instead faced general disillusionment. There were simply too many combinations; attempting to try them all was condemned to failure from the outset. No one, neither young guns nor old hands came up with a solution that worked for more than two and a half days.

Where to look for help? The Internet was still in its infancy in 1996, the web in particular was completely useless, because it mainly consisted of "homepages" that dazzled users with BLINK tags and "Under Construction" signs. However, Usenet [1] already existed and was mainly frequented by university staff, so I set about trying to describe the problem on the newsgroup rec.puzzles and asking the community for help (Figure 2).

Figure 2: The Usenet article from 1996 that started it all.

Antique Internet Helps

A few days later, the reply came from a mathematician called Barry Wolk from the University of Manitoba in Canada. He had solved the problem using the heavy weapons of discrete mathematics: a Galois field [2] with nine elements and nine mutually orthogonal Latin squares [3] generated from them.

I quickly manufactured a Perl script from his crystal-clear instructions that even non-experts could understand. The Perl script spat out a solution in a split second for not just five, but nine days (Figure 3). I was the hero!

Figure 3: The discrete math approach prints out a solution for a total of nine days within a split second.

The Canadian mathematician proposed constructing a set of Latin squares. Even the grumpy 18th century Austrian mathematician Leonhard Euler found pleasure in these structures, which are reminiscent of Sudoku puzzles containing the numbers from 1 to n exactly once for each column or row.

Figure 4 shows a Latin square with 9x9 dimensions with the numbers 1 to 9. Anyone navigating through the columns and rows will easily see that not a single number ever repeats either vertically or horizontally. Back then, Euler filled his squares with Latin capital letters A, B, … , instead of numbers; this is how the "Latin" name came about.

Figure 4: Example of a Latin square, here 9x9.

Monster Squares

However, another dimension is still missing for putting students into groups on multiple days. For this, you need a total of n-1 of these Latin squares which are orthogonal to one another, which means that the generated combinations are all unique if you compute a monster square from the superpositions of the x/y values of all the squares.

Figure 5, for example, shows four Mutually Orthogonal Latin Squares (MOLS) of the dimension 5x5 [4]. Only the blue marked values are of importance now; I'll explain the black values along the sides in a minute.

Figure 5: Because 5 is a prime number, four MOLS are quickly generated.

All four squares grouped element by element result in the megasquare in Figure 6, whose entries each consist of the corresponding values of the four original squares in the same positions. The values 5, 4, 3, and 2 are in the fourth row of the first column of the original MOLS, for example, which is why the value 5432 is in the same place in the superimposed square.

Figure 6: The four MOLS from Figure 5 return unique combinations when grouped.