the black and white sock problem

https://www.youtube.com/watch?v=GuqcnrtdUCc

B : #black socks, whole number
W : #white socks, whole number
total number of socks 200 < B + W < 250
more black than white pairs : B > W + 1
after one sock is lost, picking 2 random socks results in equal
chances to obtain a matching pair:
0.5 = P(B,B+W) * P(B-1,B+W-1) + P(W,B+W) * P(W-1,B+W-1)
P(any of A, SUM(A, B)) = A * 1 / (A+B), thus
0.5 = B/(B+W)*(B-1)/(B+W-1) + W/(B+W)*(W-1)/(B+W-1)
0.5 = B*(B-1)/((B+W)*(B+W-1)) + W*(W-1)/((B+W)*(B+W-1))
0.5 = (B²+W²-B-W)/(B²+W²+2BW-B-W)
0.5 = (B²+W²+2BW-B-W -2BW)/(B²+W²+2BW-B-W)
0.5 = 1 -2BW/(B²+W²+2BW-B-W)
-0.5 = -2BW/(B²+W²+2BW-B-W)
1 = 4BW/(B²+W²+2BW-B-W)
B²+W²+2BW-B-W = 4BW
B²+W²-2BW-B-W = 0
(B-W)^2 = B+W
now |B-W| = sqrt(B+W) with B,W in N (whole numbers) requires sqrt() =
...
= 14² = 196
= 15² = 225
= 16² = 256
...
only B+W = 15² satisfies 200 < B+W < 250, thus
(B-W)^2 = 225
B-W = +/- 15
given: B > W -> B = W + 15
now just solve for B, W
(B+W = 225): 2W + 15 = 225
W = 105
B = 120
W odd : the white sock got lost in the washing machine!