What is the Kirkman Schoolgirl Problem?

In 1850, Kirkman of a district in the Church of England asked an interesting question: a female teacher took her 15 female students for a walk every afternoon. She divided the students into 5 groups of 3 students and asked how they could arrange so that every 2 students would be in the same group one day during the week.

Kirkman himself gave an answer to this question the following year. However, this is only the case of n 15, and when n is an arbitrarily divisible positive integer, the sufficient conditions necessary for the above group to be achieved have not been proved. This is a question of existential sufficiency and necessity for combinatorial design, which has not been solved for more than 100 years.

In honor of Kirkman, a self-taught scholar in mathematical research, this famous mathematical problem is known as the "Kirkman Girl Problem".

Say that there are 15 girls in a dormitory, and they are divided into five groups to walk every night, and each girl is asked to take a walk with all the other girls once a week. It's interesting, in fact, the answer is already there. , but it was a sensation in the mathematical community for a long time.

What is its underlying meaning? Expand with changes in headcount and grouping?

Say a certain dormitory 15

per night. Five groups of walks, each of the week. All. That.

Place. Scatter. Step. Meaning, in fact, the answer has been, ever. Number. Boom.

Space. Deep implications.

Number. Group. Variation. Circumstance. Extend.

Combinatorics is related to the Latin side, and it's in any combinatorics book.

In 1850, Kirkman published an article entitled "Diary of a Lady and a Gentleman"."Question six"15 female students were asked the following questions: A female teacher led 15 girls in her class to go for a walk every day, and she divided the girls into groups of 3 into 5 groups, and asked if she could make a group plan for walking for 7 consecutive days, so that any two girls were divided into one group and only one group, that is, if you pick 2 out of 15 people, they must meet once in a group of 35 groups in a week, and only once.

It wasn't very difficult to solve the problem, Gloria first gave an answer, and then Kirkman published his own answer, which of course he already knew when he asked the question. Sylvester (also studied this issue, and later argued with Cochman about who first thought of it.)

In the same publication, Cochman published his own answer as follows (1 to 15 for 15 girls):

Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, this solution is a 15th-order Kirkman ternary where v=15, k=3, =1. Cochman not only solved the problem of the existence of the Steiner ternatry, but also gave a 2- design with parameters v=r2+r+1, k=r+1, =1 for each prime value of r, which is now called the finite projective plane. He used cyclic difference to construct projective planes of r=4 and r=8, and also found 3- designs with parameters v=2n, k=4, =1 and several other special designs.

It can be said that Kirkman is the father of composition design. The promotion of the problem is too much·· I recommend you to check it out