Kogorou wrote:1 4 9 16 25 36 49 64 81 100
Only the doors with square numbers can be open
Correct.
[spoiler=Solution]Listing the visits for each door would be helpful.
Door: Visited on trip(s)
1: 1
2: 1, 2
3: 1, 3
4: 1, 2, 4
5: 1, 5
6: 1, 2, 3, 6
7: 1, 7
8: 1, 2, 4, 8
9: 1, 3, 9
It can be inferred that the visits are actually the divisors of the number of each door.
An even total number of visits (divisors) is a closed door, while an odd one is an open door.
The only numbers that have an odd number of divisors are perfect squares. To support this, here's an example:
15:
1,
3,
5,
1516:
1,
2,
4,
8,
16Note that a particular pair of divisors multiplied to each other results in the number (1x15, 3x5, 1x16, 2x8).
But perfect squares include one divisor multiplied by itself (4x4), which means that the total number of divisors for perfect squares will always be odd.
Forgive me for the long (and probably confusing) explanation. Every math problem deserves a sound proof. ;)[/spoiler]
Kogorou, your turn.
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[/spoiler]
I missed this thread too. 
