A new one for n=9: (1*2*4*8)/(5*6*9+7) = 192/277 = 0.(693140794223826714801444043321299638989169675090252707581‌​227436823104) giving an error of 6.386336118602287E-6 The only possible better solution now would be to achieve 253/365, which I am attempting now

Some discoveries for n = 9: [(7*8)+(9*1)-4]/[6*5*3-2] = 61/88 = 0.693(18) which gives an error of: 3.4637621872946056E-5 [2*9*7+(5*1)]/[6*8*4-3] = 131/189 = 0.(693121) which gives an error of: 2.54874382521475E-5

I feel like the wording of "What is the minimum number of turns required for the black knight to ensure that you have no legal move to make?" should imply that it should work for any line that white attempts to make. I feel like that should either be reworded or we should be assuming optimal play from white.

Ah I see my problem. I incorrectly judged that at the nearest point not intersecting the cross, it would be N/5 from the center, when it's in reality (sqrt(2)/2)(N/5) since it's N/5 from each edge

Just to make sure, when the small circle is placed inside the large circle, this means that it is fully contained, not just that the center is within the circle?

This also checks out with Hint 3 since rot13(Rqhneqb, Senaxyla, naq Znggrj'f svefg yrggref pbzr orsber gurve ynfg yrggref naq gurl unir ab orneqf, gurersber vg jbhyqa'g or rabhtu whfg gb funir gurve urnq, nf gurl'q nyfb unir gb tebj n orneq)

This is about the same answer I got, the only problem is that there's no case in either of the other columns in which neither of the lines forming the square is present, meanwhile in the second column the bottom edge is in neither of the blocks. Therefore there's no way of telling whether that edge would be rejected or not.

Also, does the fact that there are 36 students imply that we are expected to produce 18 sets?

Can we assume an unlimited supply of playing cards? or just the 12 provided in the picture?

Could I recommend shortening the hour hands on these? It's a bit ambiguous looking on some of them and I want to make sure I have it right.