Counting Squares in Bigger Squares

Date: 02/29/2000 at 22:45:51
From: Julz
Subject: Edge

I have a math question that reads:

How many Edge 2's can make up Edge 4? In other words, how many times 
does a square that is edge 2 (a four unit square - two by two) fit 
into a square that is edge 4 (a 16 unit square - four by four). My 
math teacher says the answer is 8.

Is this sort of like the game tetris? Like how many different shapes 
in a 16 unit box can I make with just 4 units?

Sincerely, 
Julz

Date: 03/09/2000 at 14:27:44
From: Doctor Twe
Subject: Re: Edge

Hi Julz - thanks for writing to Dr. Math.

Earlier another student wrote with a similar question; hers involved 
an entire checkerboard. Here's what I wrote to her. The reasoning for 
the answer to your question (which should be 9, not 8) is contained in 
it.

Question: How many squares on a checkboard?

Answer:

A checkerboard - which is the same as a chessboard - is an 8x8 
arrangement of squares, so it has 64 small squares. Of course, you 
could make a "square" comprised of a 2x2 block of the small squares, 
or a 3x3 block, etc. If you count all of these "squares," you would 
get a much larger number. To count these larger squares, it is easiest 
to figure out which points on the checkerboard can be the upper left 
corner of a 2x2 square and count them, then do the same for 3x3, 4x4, 
etc. When you add up all of these values, you have the total number of 
"squares" in a checkerboard. Let's look at a smaller example - a 4x4 
"checkerboard." I'll label the points on the board like this:

     A---B---C---D---E
     |   |   |   |   |
     F---G---H---I---J
     |   |   |   |   |
     K---L---M---N---O
     |   |   |   |   |
     P---Q---R---S---T
     |   |   |   |   |
     U---V---W---X---Y

There are 16 squares of size 1x1.

Points A, B, C, F, G, H, K, L and M can be the upper left corners of 
2x2 squares, for a total of 9. Note that points D, E, I, J, N and O 
can't be upper left corners because we can't go 2 squares to the right 
of these points. Likewise, points P through Y can't be upper left 
corners because we can't go 2 squares down from these points.

Only points A, B, F and G can be the upper left corners of 3x3 
squares, for a total of 4. (The reasons the others can't be are 
similar to the reasons given above.)

Finally, there is only one 4x4 square, with point A being its upper 
left corner. So we have 16 + 9 + 4 + 1 = 30 total squares.

Do you see a pattern in the numbers we were adding in the last step? 
Using that pattern, we can determine the total number of squares on an 
8x8 checkerboard without counting corners.


As to the relation of this to the game tetris, tetris is a little more 
complicated. Not only do you have to consider different shapes (I'd 
consider each one separately), but you also have to consider rotations 
and reflections of the shapes. For example, the shape:

     +---+---+---+
     |   |   |   |
     +---+---+---+
     |   |
     +---+

can be rotated to three other positions:

     +---+                           +---+---+
     |   |                           |   |   |
     +---+                 +---+     +---+---+
     |   |                 |   |         |   |
     +---+---+     +---+---+---+         +---+
     |   |   |     |   |   |   |         |   |
     +---+---+     +---+---+---+         +---+

and can be reflected and rotated to four other positions:

     +---+---+---+         +---+                       +---+---+
     |   |   |   |         |   |                       |   |   |
     +---+---+---+         +---+     +---+             +---+---+
             |   |         |   |     |   |             |   |
             +---+     +---+---+     +---+---+---+     +---+
                       |   |   |     |   |   |   |     |   |
                       +---+---+     +---+---+---+     +---+

If we count every way that each of these rotations and reflections 
will fit in a 4x4 square, we'll get a much larger number.

I hope this helps. If you have any more questions, write back.

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