Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2004 MATH025.1001
Invariants
Usually when one tries to prove that something is possible, one proves it by
constructing a solution.
This is not always the case, but it is a fairly common rule.
How do we prove that something is impossible?
One way is by finding invariants.
Below we give an example.
-
Suppose we have a 5 by 5 checkers board
We want to cover this board with red dominoes.
We have a large stack of them.
Each domino stone is exactly the size of two fields of the board.
Here are pictures of some example dominoes:
- Question: Can we exactly cover the 5 by 5 board with
non-overlapping dominoes?
- Answer: No.
Each domino covers 2 fields.
So each bunch of dominoes covers an even number of fields.
That is the invariant.
However, the 5 by 5 board has 25 fields, which is an odd number.
-
The following is a preliminary step towards a less trivial example.
Suppose we have an 8 by 8 board
We also have 32 red dominoes.
Each domino stone is exactly the size of two fields of the board.
Here are two of them:
- First a simple Question: Can we exactly cover the 8 by 8 board
with 32 dominoes?
- Answer: Yes.
Simply line them all up horizontally.
Here is another way: Line them all up vertically.
There are many other ways.
For example, line them up horizontally on the top half of the board, and line
them up vertically on the bottom half of the board.
-
Now let us make it less trivial.
Suppose we have an 8 by 8 board as above, except that we cut off the bottom
left field and the top right field:
We also throw away one of the 32 dominoes.
- Question: Can we exactly cover the mutilated board with 31
dominoes?
- Answer: No.
This answer is not so obvious, since the mutilated board has 62 fields, which
equals 31 times 2.
So the earlier invariant, which uses that bunches of dominoes cover even
numbers of fields, no longer suffices.
The proof below involves finding another invariant.
Here is how.
Color alternating fields like on a chess board:
Each domino covers exactly one green and one blue field.
So each bunch of dominoes covers an equal number of green and blue fields.
That is the new invariant.
The mutilated board has 32 green fields, and has 30 blue fields.
So we can not cover the mutilated board with 31 dominoes.
In class we discussed the other `invariant' case involving left-over angles,
of solids like the platonic solids.
I know that I disappoint you all with this, but I decided that I will not
discuss that example on the exam.
Last updated: December 2004
Comments & suggestions:
wimr@mscs.mu.edu