At a boarding school, in the evening, 9 students always take walks in groups of 3. How can the groups be arranged so that each person walks in a group with every other person exactly once in 4 days?
I would like to know whether there is a trick to these type of questions. My workings are students numbered 1-9 and I did 123 124 125 126 127 128 129. And I would also like to know whether my answer could be correct, 9 students, 3 spots is 27 different combinations?
To solve this problem, you need to find a way to arrange the groups such that each student walks with every other student exactly once in 4 days.
One approach to working on this type of problem is to use the concept of graph theory. In this case, we can represent the students as vertices in a graph, and the walks between each pair of students as edges.
To start, let's label the students as A, B, C, D, E, F, G, H, and I.
One possible way to arrange the groups over 4 days is as follows:
Day 1:
Group 1: A, B, C
Group 2: D, E, F
Group 3: G, H, I
Day 2:
Group 1: A, D, G
Group 2: B, E, H
Group 3: C, F, I
Day 3:
Group 1: A, E, I
Group 2: B, F, G
Group 3: C, D, H
Day 4:
Group 1: A, F, H
Group 2: B, D, I
Group 3: C, E, G
In this arrangement, each student walks with every other student exactly once in the four days.
Now, let's address your question about whether your answer is correct. You mentioned listing all the combinations, resulting in 27 different groups. Unfortunately, this approach is not correct because it violates the condition of each person walking with every other person exactly once in 4 days. It is important to ensure that each student walks with every other student exactly once in the given number of days.
To solve this type of problem, you can use strategies from graph theory, as demonstrated above, or you can use other mathematical techniques like combinatorics or permutation theory. There might be various valid arrangements depending on the problem, so exploring different approaches is key.
In summary, to answer this specific problem, you need to find a way to arrange the groups so that each student walks with every other student exactly once in 4 days. Using graph theory, we can represent the students as vertices and the walks as edges. By carefully arranging the groups, you can find a valid solution to the problem.