CombinatoricsDifficulty 5.4AIME, harderProve itBaltic Way
In the very large Baltic-Way-City (in the far future) there are 16 hospitals. Every night exactly 4 of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after 20 days every pair of hospitals were on duty exactly once? If not, prove the non-existence. If yes, give a schedule.
Solution
The answer is yes. Let the hospitals be numbered 1,2,…,16. The hospitals on duty are the 4 on the rows.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
5
9
13
2
8
10
15
3
6
11
16
4
7
12
14
1
6
10
14
2
7
9
16
3
5
12
15
4
8
11
13
1
7
11
15
2
6
12
13
3
8
9
14
4
5
10
16
1
8
12
16
2
5
11
14
3
7
10
13
4
6
9
15
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