Kirkman’s Schoolgirls - Lycée Emmanuel d'Alzon (Nîmes)

Fifteen young schoolgirls walk every day of the week, from Monday to Sunday, in an orderly way, forming five rows of three schoolgirls each. How should we organize them every day of the week so that no pair of schoolgirls shares the same row for more than one day?
With the simplify version
First, we can simplify the topic by considering nine schoolgirls. If we perform a calculation based on the set {9;8} (where 9 represents all the girls, and 8 represents all the girls excluding one), we can make the simplified calculation: 8÷2=4. Therefore, we can suppose that the nine girls can walk for four different days without being next to the same neighbors. Then, if we represent this scenario using tables, the result is also four days.

Then, we can deduce that nine schoolgirls can walk for four days under these conditions.
With fifteen girls
Afterward, we can apply the same reasoning to fifteen schoolgirls. If we perform the calculation based on the set {15;14}, we find: 14÷2=7. Therefore, we can suppose that the fifteen girls can walk for seven different days without being next to the same person.
And, if we model this with circles and triangles that constitute the circle, the result is also seven. Consequently, we can deduce that fifteen schoolgirls can walk for seven days.
To solve this problem, we can also use a matrix method with the Sarrus rules and we find the same previous result.