The Locker Problem


1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
3
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
4
1
0
0
1
1
1
1
1
0
0
1
0
1
0
0
1
1
1
1
1
5
1
0
0
1
0
1
1
1
0
1
1
0
1
0
1
1
1
1
1
0
6
1
0
0
1
0
0
1
1
0
1
1
1
1
0
1
1
1
0
1
0
7
1
0
0
1
0
0
0
1
0
1
1
1
1
1
1
1
1
0
1
0
8
1
0
0
1
0
0
0
0
0
1
1
1
1
1
1
0
1
0
1
0
9
1
0
0
1
0
0
0
0
1
1
1
1
1
1
1
0
1
1
1
0
10
1
0
0
1
0
0
0
0
1
0
1
1
1
1
1
0
1
1
1
1
11
1
0
0
1
0
0
0
0
1
0
0
1
1
1
1
0
1
1
1
1
12
1
0
0
1
0
0
0
0
1
0
0
0
1
1
1
0
1
1
1
1
13
1
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
1
1
1
1
14
1
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
1
1
1
1
15
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
16
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
1
1
1
1
17
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
1
18
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
1
1
19
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
20
1
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0


If we denote 0 as open and 1 as close, then we can see the pattern that the doors numbered in perfect squares - such as 1, 4, 9, 16, 25, 36, 49, ..., 961 - are close and the rest are open.

Therefore, the doors that are closed are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961. The rest are open.

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