Seven dwarfs are sitting at a round table.
Each has a cup, and some cups contain milk.
Each dwarf in turn pours all his milk into the other six cups, dividing it equally among them.

After the seventh dwarf has done this, they find that each cup again contains its initial quantity of milk.

How much milk does each cup contain, if there were 42 ounces of milk altogether?

Source: 1977 All Soviet Union Math Olympiad Problem

In the following table, the dwarfs are lettered a through g. Each row shows what at the end his own share consists of what fraction of the original share of a, of b etc. is present in his cup.

       of a's         of b's    of c's     of d's   of e's  of f's
a   70993/279936  16807/46656  2401/7776  343/1296  49/216  7/36  0
b   63217/279936  9031/46656  2401/7776  343/1296  49/216  7/36  0
c   54145/279936  7735/46656  1105/7776  343/1296  49/216  7/36  0
d   43561/279936  6223/46656  889/7776  127/1296  49/216  7/36  0
e   31213/279936  4459/46656  637/7776  91/1296  13/216  7/36  0
f   16807/279936  2401/46656  343/7776  49/1296  7/216  1/36  0
g   0  0  0  0  0  0  0

For example a has 70993/279936 of his own original contents plus 16807/46656 of b's original contents plus 2401/7776 of c's original contents, etc.

As a through f now contain equal amounts (7 ounces):

70993/279936 *a + 16807/46656 *b + 2401/7776 *c + 343/1296 * d + 49/216 * e + 7/36 * f = 7
etc.

where the variables a through f represent the original amounts they held.

Solve the six equations in six unknowns.

Table generated by:

    5   dim Ofa(7),Ofb(7),Ofc(7),Ofd(7),Ofe(7),Off(7),Ofg(7)
   10   Ofa(1)=1
   20   Ofb(2)=1
   30   Ofc(3)=1
   40   Ofd(4)=1
   50   Ofe(5)=1
   60   Off(6)=1
   70   for Gen=1 to 7
   80     Splitofa=Ofa(Gen)//6
   90     Splitofb=Ofb(Gen)//6
  100     Splitofc=Ofc(Gen)//6
  110     Splitofd=Ofd(Gen)//6
  120     Splitofe=Ofe(Gen)//6
  130     Splitoff=Off(Gen)//6
  140     Splitofg=Ofg(Gen)//6
  150     for Rcp=1 to 7
  160       if Rcp<>Gen then
  170         :Ofa(Rcp)=Ofa(Rcp)+Splitofa
  180         :Ofb(Rcp)=Ofb(Rcp)+Splitofb
  190         :Ofc(Rcp)=Ofc(Rcp)+Splitofc
  200         :Ofd(Rcp)=Ofd(Rcp)+Splitofd
  210         :Ofe(Rcp)=Ofe(Rcp)+Splitofe
  220         :Off(Rcp)=Off(Rcp)+Splitoff
  230         :Ofg(Rcp)=Ofg(Rcp)+Splitofg
  240       :endif
  250     next
  260     Ofa(Gen)=0
  270     Ofb(Gen)=0
  280     Ofc(Gen)=0
  290     Ofd(Gen)=0
  300     Ofe(Gen)=0
  310     Off(Gen)=0
  320     Ofg(Gen)=0
  330     print Gen
  340     for I=1 to 7
  350      print Ofa(I);Ofb(I);Ofc(I);Ofd(I);Ofe(I);Off(I);Ofg(I)
  360     next
  370   next Gen
  380   for I=1 to 7
  382      Tota=Tota+Ofa(I):Totb=Totb+Ofb(I):Totc=Totc+Ofc(I):Totd=Totd+Ofd(I):Tote=Tote+Ofe(I):Totf=Totf+Off(I):Totg=Totg+Ofg(I)
  385   next
  400   print Tota;Totb;Totc;Totd;Tote;Totf;Totg

 

Posted by Charlie on 2013-10-23 14:55:44