May 24, 2004
Problem of the Month

Milk Problem
Source: Mathematical Puzzles
of Sam Loyd, 1960

Two Sherando students each asked Honest John Handley for two quarts of milk.

One student had a five-quart pail and the other had a four-quart pail.

John had only two ten-quart cans, each full of milk.

How did he measure out exactly two quarts of milk, one for each student?
And what's the smallest number of pourings needed?

Solution to the Problem:

It can be done in 9 moves in several different ways:

move ten ten five four   
  1  10   5   5    0     
  2  10   5   1    4    
  3  10   9   1    0     
  4   6   9   1    4    
  5   7   9   0    4    
  6   7   9   4    0    
  7   7   8   5    0    
  8   7   8   1    4     
  9  10   8   1    1      

move ten ten five four   
  1  10   5   5    0     
  2  10   5   1    4    
  3  10   9   1    0     
  4  10   9   0    1    
  5   7   9   0    4    
  6   7   9   4    0    
  7   7   8   5    0    
  8   7   8   1    4     
  9  10   8   1    1      

move ten ten five four   
  1  10   6   0    4     
  2   5   6   5    4    
  3   9   6   5    0     
  4   9   6   1    4    
  5   9   7   0    4    
  6   4   7   5    4    
  7   8   7   5    0
  8   8   7   1    4     
  9   8  10   1    1      

move ten ten five four   
  1  10   6   0    4     
  2   5   6   5    4    
  3   9   6   5    0     
  4   9   2   5    4    
  5   9   7   0    4    
  6   4   7   5    4    
  7   8   7   5    0
  8   8   7   1    4     
  9   8  10   1    1      

Mikael Wetterholm stated that it was easy to figure out the last three pourings but then it took a few tries to solve the rest of the problem.   He found fifteen different ways to measure out two quarts of milk (one for each student) with only nine pourings.

Correctly solved by:

1. Jeffrey Gaither	Winchester, Virginia	9 moves -- did not waste milk
2. Alex Truong	Harrisonburg, Virginia	9 moves -- did not waste milk
3. Celeste Doerwaldt	Winchester, Virginia	9 moves -- did not waste milk
4. Mikael Wetterholm	Danderyd, Sweden	9 moves -- did not waste milk
5. Bella Patel	Harrisonburg, Virginia	9 moves -- did not waste milk
6. Helna Patel	Harrisonburg, Virginia	9 moves -- did not waste milk
7. Dave Smith	Toledo, Ohio	10 moves -- did not waste milk
8. Walt Arrison	Philadelphia, Pennsylvania	10 moves -- wasted 18 quarts of milk
9. James Alarie	University of Michigan -- Flint Flint, Michigan	11 moves -- did not waste milk

Send any comments or questions to: David Pleacher