In 1850, an English reverend published a problem that became one of the classics of recreational mathematics. ‘It looks like a puzzle, an enigma, but behind it there are very profound aspects’, says scholar.

Imagine that your long-awaited pay raise depends on you doing just one thing.

Your boss is organizing a conference and has confirmed the presence of seven leading specialists who will debate among themselves in round tables. And it asks you that each table only have three debaters.

So far, so good, right? Already viewing your bank account?

But, during the conversation with the debaters, you discover one detail: they are the best in their areas, but they don’t get along with each other — and each one, in their own way, imposes a condition:

“I can attend tables as needed, but I want to be present with each of the other six guests only once, no more, no less.”

It seems difficult, but don’t despair.

What your boss is asking is very similar to the question posed by British mathematician Thomas Kirkman in 1850—known as the schoolgirl problem.

With the help of mathematics professor Raúl Ibáñez, from the University of the Basque Country in Spain, we’ll tell you what it’s all about.

“The problem of schoolgirls has fascinated people for a long time. It looks like a puzzle, an enigma, but it has very profound aspects behind it”, says Ibáñez, who is a science popularizer and author of several books and articles on mathematics. One of his books devotes a chapter to this problem.

“It seems easy, but it is intrinsically very complicated, and its resolution is not always simple”, says the professor.

## group theory

Kirkman was born in Manchester, England, in 1806.

A teacher remarked at school that he had the potential to be accepted to Cambridge University, but his father had other plans.

“Thomas was forced to leave school at the age of 14 and go to work in his father’s office,” as professors John Joseph O’Connor and Edmund Frederick Robertson of the University of St. Andrews in the UK.

“After nine years working in the office, Thomas went against his father’s wishes and entered Trinity College in Dublin, Ireland, to study mathematics, philosophy, the classics and science in order to obtain a degree”, say the professors.

In 1835, Kirkman returned to England, and four years later he became vicar of an Anglican parish, holding the position for 52 years. He got married and had three children.

As Robin Wilson, Emeritus Professor of Pure Mathematics at the Open University in the United Kingdom, pointed out in the article* The Early History of Block Designs* (“The Early History of Block Drawings”, in free translation), Kirkman’s parochial duties “occupied little of his time”.

Therefore, the reverend “concentrated a lot of efforts on his mathematical research, especially on topics of algebra and combinatorics”.

## the triple systems

In 1846, Kirkman presented his first article, with the title *On a Problem in Combinations* (“About a problem of combinations”, in free translation). It was published in 1847 in the Cambridge and Dublin Mathematical Journal.

The article is considered a pioneer of Steiner’s triple system, several years before its presentation by the Swiss geometer Jakob Steiner (1796-1863), considered one of the most important of the 19th century.

“Perhaps these triple systems should have been called Kirkman’s systems, since he was the first to publish them”, says Ibáñez.

Throughout his career, Kirkman delved deeper into group theory and made important contributions to combinatorics.

## recreational math

Kirkman published the problem of schoolgirls in The Lady’s and Gentleman’s Diary, devoted to mathematical questions, riddles and poetry.

It was a puzzle, a mathematical recreation, laid out this way:

“Fifteen young students go out for a walk every day of the week, from Monday to Sunday, in an orderly fashion, forming five lines of three schoolgirls each. How should we organize them every day of the week so that no pair of schoolgirls share the same queue for more than a day?”

This approach caught the attention of several well-known mathematicians, including the British Arthur Cayley (1821-1895), who quickly published a solution. Kirkman would introduce another, and from then on, several resolutions would emerge.

Kirkman came up with the schoolgirl problem just as he was writing his article on triple systems.

“We have n elements, 1, 2, 3 up to n, and the idea is to create collections of three numbers from this set, called blocks, so that each pair of elements appears exactly in a trio”, explains Ibáñez.

What Kirkman asks in his problem is that, for 15 people or elements, we can develop a triple system separated into seven groups (one for each day of the week), so that, in each of them, are all the elements — in this case , the schoolgirls.

## Room’s squares

Cayley is considered one of the founders of the British school of pure mathematics, which emerged in the 19th century.

In 1850, he decided to pay attention to the 15 schoolgirls problem and came up with a solution using what are now known as Room squares — which would later be documented by the Australian mathematician Thomas Gerald Room (1902-1986).

The professor explains that, in a square of Room, we have n+1 symbols.

Imagine 8 numbers, from 1 to 8.

Since we chose eight symbols, we make a 7×7 table: seven rows and seven columns.

But three conditions must be met:

- Each square is empty or has a pair of numbers. For example, one square might have 35, another might have 86, another might have 13 or nothing at all.
- Each symbol appears only once in each row and each column. If we take a line, for example, the 1 will appear in one of the squares, the 2 in another and so on until the 8. In the columns, the same thing happens, but they will appear forming a pair of numbers.
- Each unordered pair of symbols appears only once. The par 12, for example, appears only once in the entire table, the 13 appears only once, and so on until the end, until the par 78.

An example would be this:

What Cayley did was use this kind of Room square and combine it with the triple systems Kirkman was already studying to come up with a solution to the schoolgirl problem.

Cayley distributed the 15 students as follows: he indicated the first seven with letters from “a” through “g”, and the other eight with numbers, from 1 to 8.

The numbers serve to form a square of Room, as illustrated above, and the letters to make triple systems of order seven, like this one:

These trios are placed to the left of Room’s square, like this:

## The solution

From that framework, a solution emerges.

Let’s transpose the framework to the 15 schoolgirls and the seven days they go out for a walk.

But first, let’s name the letters and numbers in the Cayley table:

- a=Ana
- b=Bia
- c=Carol
- d=Diana
- e=Emma
- f=Fany
- g=Ginny
- 1=Mary
- 2=Katy
- 3=Yeny
- 4=Lola
- 5=Sofia
- 6=Gabi
- 7=Pili
- 8=Yoli

The solution comes from the square of letters and numbers above. Each line in this square gives us the groups of three students for each of the seven days of the week.

That is, on Monday it is abc, d35, e17, f82 and g64. The solution, with our students, would be this:

## The art of combinatorial analysis

Both Kirkman and Cayley “knew there was something deep behind this problem, so they tackled it.”

“Combinatorial analysis is the art of selecting or ordering the elements of a certain set” — and this is precisely what Cayley shows us with his solution. The schoolgirls’ problem is one of organization.

“Students and how to group them to go to school each day is a metaphor for a mathematical structure, actually combinatorial, which can be used in many other aspects of our lives”, says Ibáñez.

“This is why mathematics is abstract — so that it is a tool that can be used in very different contexts, such as physics, biology, chemistry or medicine.”

According to him, the mathematics involved in the schoolgirls’ problem is part of a whole branch that is fundamental in the theory of codes and cryptography, planning, geometry, design of statistical experiments, theory of computation and communication networks.

“Everything that comes from trying to solve a puzzle ended up being converted into two mathematical theories: Steiner’s triple systems and the theory of block design, both of which have many practical applications”, says Ibáñez.

This happens because mathematics “is not satisfied” with solving the problem.

“In some cases, like this one, it also looks at how many distinct forms of solution there are. And for Kirkman’s schoolgirl problem, it was shown in the early 20th century that there were 80 distinct solutions.”

## The problem that causes the most problems

Schoolgirls also raised new problems.

“Another common practice in Pythagoras’ science is to expose the problem in a more general way”, says Ibáñez.

“For this reason, the problem of schoolgirls was proposed for groups with other numbers of students.”

The solution for all cases did not arrive until 1968, when mathematicians Ray-Chaudhuri and RM Wilson published the “complete solution for the general case”.

Still, the problem remains open, as Steiner’s triple systems or, more generally, block drawing is a “very active” branch of mathematics.

“A puzzle like this one, which, in principle, was a small question, has become a theory with hundreds of open problems, research, articles and books”, says Ibáñez.

And while trying to solve it, many mathematicians used and developed different techniques.

The American mathematician Martin Gardner, for example, published in Scientific American magazine a geometric solution to the problem of schoolgirls: a circle, with numbers and triangles on it, which offers a different answer as it is rotated.

Returning to the question that could give you the desired salary increase, the answer is to assign a number to each of the guests and create a triple system, which will lead, for example, to seven tables:

And if you want to thank anyone for a well-deserved pay raise, thanks undoubtedly go to Thomas Kirkman and, of course, Professor Ibáñez.

🇧🇷*Graphics**: Manuella Bonomi and **Ana Lucia Gonzalez*

This text was originally published at https://www.bbc.com/portuguese/geral-63711028