Mod and Latin Squares!!!?

latin squares are orthogonal if no two cells contain the same ordered pair. I think a good way to start might be...

prove that if

(f_j , g_j) = (f_k , g_k) then j must equal k.

Things usually work best by contradiction. Assume j not equal k and use the congruences to obtain a contradiction. I'll look at it some more.

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As I guessed

Say that (f_j , g_j) = (c,d) = (f_k , g_k)
So we know that for some x1,x2,y1,y2

c = x1 + y1 (mod n)
c = x2 + y2 (mod n)
d = 2*x1 + y1 (mod n)
d = 2*x2 + y2 (mod n)

c - x1 = y1 (mod n)
c - x2 = y2 (mod n)
Which implies
d = 2*x1 + (c - x1) (mod n)
d = 2*x2 + (c - x2) (mod n)
d = x1 + c (mod n)
d = x2 + c (mod n)

x1 = x2 (mod n)
y1 = y2 (mod n)

So if the ordered pair is the same the x,y are the same (mod n). Which for latin squares are all the values we worry about.