If there are two people in the room, A and B, the odds that they do NOT share the same birthday is 364/365 (I'm ignoring February 29th)
If there are three, A,B, and C, we have three pairs to consider, AB, AC, and BC, so the odds that no two share a birthday becomes (364/365)^3
If there are four, we have AB, AC, AD, BC, BD, and CD = six pairings, so the odds are (364/365)^6 that they do not share a birthday.
Since the exponents are 1, 3, and 6, generalizing, the odds than N people have no pair with identical birthdays becomes (364/365)^n, where n = N+ (N-1) + (N-2) ... 2 + 1
So, the question is, for what n is (364/365)^n closest to 0.5, and what N corresponds to that n. Here's where it gets shaky for me. I think the next step needs to be
log 0.5 = n(log (364/356))
-0.30102999566 = n(0.00965138567)
so n = -31.1903394966, but that seems like nonsense, so my solution would be to plug in values for n and find the one closest to 0.5
(364/365)^20= 0.94660846509
(364/365)^50 = 0.87181825733
(364/365)^100= 0.76006707381
(364/365)^200= 0.57770195669
(364/365)^250=0.50365111314
(364/365)^253= 0.49952284596
So n is about 252-253. N must be 1+2+3 ... + N until we get close to 252
For N=10, n is 1+2+3+4+5+6+7+8+9+10=55
N=15, n= 55+11+12+13+14+15= 120
N=20, n= 120+16+17+18+19+20 = 210
N=23, n=210+21+22= 253
I've seen the answer and know that it is correct, but I can't derive it mathematically.