25 golfers have a tournament. They split up into 5 groups each with 5 players. After 6 playing rounds, every person has played against all of the others. How was the 25-5-6 tournament arranged? I don't see any clever ways of solving it. 1: abcde, fghij, klmno, pqrst, uvwxy 2: afkpu, bglqv, chmrw, dinsx, ejoty 3: agmsy, bhntu, ciopv, djkqw, eflrx 4: ahoqx, bikry, cjlsu, dfmtv, egnpw 5: ailtw, bjmpx, cfnqy, dgoru, ehksv 6: ajnrv, bfosw, cgktx, dhlpy, eimqu --Ed Pegg Jr, www.mathpuzzle.com ---------------------------------------------------------------------------- Dear Ed, There is a simple mathematical way of constructing the structure you want. Let the 25 peope be represented by the points (x,y) where each of x and y is an integer from 0 to 4 inclusive. We use arithmetic modulo 5. There are six families of parallel lines: (1) Slope 0: y=0, y=1, y=2, y=3, y=4. (2) Slope 1: y=x, y=x+1, y=x+2, y=x+3, y=x+4. (3) Slope 2: y=2x, y=2x+1, y=2x+2, y=2x+3, y=2x+4. (4) Slope 3: y=3x, y=3x+1, y=3x+2, y=3x+3, y=3x+4. (5) Slope 4: y=4x, y=4x+1, y=4x+2, y=4x+3, y=4x+4. (6) Slope infinite: x=0, x=1, x=2, x=3, x=4. Such a structure is called a finite affine plane. You can of course go to higher dimensions. Best regards, Andy Liu. ----------------------------------------------------------------------------- label the puzzlers with pairs of integers (i, j) with 0 <= i,j < 5. on round k , group the golfers according to the residue of f_k(i,j) modulo 5 , where the linear functions are f_k(i, j) = i + k * j (for k = 1, 2, 3, 4, 5) and f_6(i, j) = j . note: this kind of construction generalizes very easily. the sets are basically lines in an affine plane over a finite field. mike reid