Milk Bottle Puzzle

Date: 09/24/2001 at 13:36:45
From: clare davies 
Subject: Logical

You have a milk crate that is 6 holes wide by 4 holes deep. You have 
18 milk bottles. You have to put an even number of bottles into every 
row and column.

Date: 09/25/2001 at 12:16:41
From: Doctor Greenie
Subject: Re: Logical

Hi, Clare -

Cool problem!  It fooled me at first.

You have 18 milk bottles to put in a crate which is 6 holes wide by 4 
holes deep; you want an even number of bottles in each row and column.  
There are 18 milk bottles and 24 holes, so there will be 6 empty 
holes. If the crate is 6 holes wide by 4 holes deep and each row and 
column is to have an even number of bottles, then each row and column 
will also have an even number of holes. Looking for arrangements of 
the 6 empty holes that satisfy the requirements is easier than looking 
for arrangements of the 18 milk bottles that satisfy those 
requirements.

Can you find an arrangement of the six empty holes that will satisfy 
the conditions of this problem? There are many such arrangements...

Here is a start on one of them:

   +---+---+---+---+---+---+
   |   |   |   |   |   |   |
   +---+---+---+---+---+---+
   |   |   | X |   |   | X |
   +---+---+---+---+---+---+
   |   |   |   |   |   |   |
   +---+---+---+---+---+---+
   |   |   | X |   |   |   |
   +---+---+---+---+---+---+

I started this with an empty hole in row 2, column 3. Then I made 
row 2, column 6 an empty hole to give me an even number of empty holes 
in row 2; and I made row 4, column 3 an empty hole to give me an even 
number of empty holes in column 3.  

This leaves you with an odd number of empty holes in row 4 and in 
column 6; you need to assign 3 more empty holes such that each row 
and column has an even number of empty holes. Here is a big hint: you 
must NOT make row 4, column 6 an empty hole; if you do, you can't 
complete the task.

When you finish assigning the 6 empty holes so that each row and 
column contains an even number of empty holes, put milk bottles in the 
remaining holes in the milk crate, and each row and column will have 
an even number of milk bottles.

I hope this helps. Thanks for the interesting problem - I enjoyed 
playing with it!

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/