[rushyrulz]

Okay, so we know that none of the doors will work if we try them at a random time, even if we have the correct door, Archie said that all of the doors have been tried at some time or another.

Incorrect doors: 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
'An evenhanded decision will lead you to your death'
So I think we can safely rule out:
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

So the remaining doors are:
9 15 21 25 27 33 35 39 45 49 51 55 57 63 65 69 75 77 81 85 87 91 93 95

"When the days equal the doors, look to the clock, for all is done"

We still need to decipher this clue.