Welcome to NRICH.

 
Bottles in milkcrate problem
By Rachelle Najman (T4009) on Tuesday, January 30, 2001 - 05:09 am:

Hi all,
I am a teacher candidate taking this nature of mathematics course at school for elementary-aged children. I am stumped with one question from a problem set. I hope you can provide me with some insight. Thanks. Here is the problem:

A certain square milkcrate can hold 36 bottles of milk. Can you arrange 14 bottles in the crate so that each row and column has an even number of bottles? (Depict the crate)

This problem was taken from "Thinking Mathematically" by John Mason (pg. 181)

By Oliver Samson (P3202) on Tuesday, January 30, 2001 - 10:03 pm:

How about:

ooooxx
ooxxxx
oxoxxx
oxxoxx
xxxxoo
xxxxoo

where x is an empty space, and o is a bottle. I think there are many arrangements where there is one row of 4, one column of four, and the rest all with 2.

By Rachelle Najman (T4009) on Wednesday, January 31, 2001 - 04:56 am:

Any other depictions or ideas?

By Emma McCaughan (Emma) on Wednesday, January 31, 2001 - 10:05 am:

Putting 12 bottles in the crate is easy to do systematically. Put one in the top left, then make the first row up to 2, then the second column up to 2, then the second row up to 2, etc, until you get to the bottom right, when you put one in the bottom left to make up column 1:
xxoooo
oxxooo
ooxxoo
oooxxo
ooooxx
xoooox
I did this, and then adapted it: I took out the bottle in the bottom left, and put that somewhere else (doesn't matter much where).
xxoooo
oxxooo
ooxxxo
oooxxo
ooooxx
ooooox
Now there are two "odd" columns and two "odd" rows, so we can see where to put the last two.

If you can follow a reasonably systematic method like this, it shouldn't be too difficult to cope with a different size crate or different number of bottles.

By Rachelle Najman (T4009) on Wednesday, January 31, 2001 - 04:54 am:

Thanks for the help.