John Conway writes:

Exactly what are these the probabilities of? I was using the probability that a given triple a,b,ab should be of form square,square,nonsquare. Are yours the conditional probability that ab should be a non-square GIVEN that a,b are squares?

[I mention that there is a third natural probability around, namely the probability that a^2.b^2 should be a nonsquare.]

I see! I first calculated the second probability, and saw that it wasn't what you were describing. Also, the denominators aren't as pretty, because they reflect the somewhat flukey number of squares. So I switched to the third, and since it agreed with your number (1/6) for A4 (and since I overlooked the 1/36 you got for Q12) I thought that was what you meant. So the numbers I reported were the probability that aabb is a nonsquare. Looking at the first probability I see it can't exceed min(s^2,1-s) where s is the density of squares--that's at most tau^-2 ~ .382, and it's even less because not every product of squares can be a nonsquare. Up to order 255, the only group that exceeds 1/6 is SmallGroup(48,50), or <a,b,c : a^3 = b^2 = c^2 = (a b)^3 = (a c)^3 = (b c)^2 = (a b a^-1 c)^2 = 1> . It has a probability of 5/24 that a,b,ab are square,square,nonsquare. Dan