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