D R. D R E W ' S M A T H 2 : P O W W R I T E - U P s
Checkerboard Squares: Anchor 7
1. Problem Description
In this POW, we are asked to investigate an
8×8
checkerboard and count the number of squares. As shown in the
figure on the left below, it is obvious that there are 64 gray and white squares (32 gray and 32 white), but these are
not they only squares that can be found on the board. In the problem statement, we were given an example of one
3×3
square that can also be found on an
8×8
checkerboard, shown in the figure on the right below. The question
were to answer is: How many squares are there on an
8
×
8
checkerboard?
2. Process Description
When I first started working on this problem, I knew that I would need to use the Stay Organized Habit of a
Mathematician; otherwise, it would be really easy to miss a square or possibly even count a square twice. I needed a
systematic way of counting (Be Systematic) but I really didn’t want to go to the trouble of doing all that counting—it
seemed as if it might be a waste of time. Instead, I decided to use the Start Small Habit of a Mathematician and
count the squares in a
1×1, 2 ×2, 3 ×3
and a
4×4
checkerboard. It was my hope that a pattern would emerge that
would allow me to determine the number of squares in an
8×8
(and justify the reasoning, of course). For reference,
here are my checkerboards:
Starting with a
1×1
checkerboard, the number of squares was trivial to count: there is one (and only one) square.
For a
2×2
checkerboard, there are
2×2=4
“
1×1
squares” and 1 “
2×2
square” (for a total of three). For a
3×3
checkerboard, there are
3×3=9
“
1×1
squares”, 4 “
2×2
squares” and 1 “
3×3
square” (for a total of 14). To
count the number of
2×2
squares, I visualized a
2×2
square that starts in the upper left and then moves one square
at a time to the right. When it hits, the end, it moves back to the left and one square down and repeats. This
movement was my way of being systematic and ensuring I was counting correctly. The diagram below illustrates:
This approach made it straightforward to count the number of squares in a
4×4
checkerboard. There are:
◼
16 “
1×1
squares”
◼
9 “
2×2
squares”
◼
4 “
3×3
squares”
◼
1 “
4×4
squares”
When adding up these numbers (to get a total of 30 squares) I noticed two patterns emerge! First, the number of
squares in a
2×2
squares is
22=4
more than in a
1×1
; the number of squares in a
3×3
square is
32=9
more than
in a
2×2
square; and the number of squares in a
4×4
square is
42=16
more than in a
3×3
square. Second, I
noticed that all the numbers I was adding up were perfect squares. I thought about this for a few minutes and then
realized that had to be the case and the method I used to count the number of squares explains why: the number of
horizontal and vertical movements of a small square in a larger square has to be the same, so the total number of
small squares has to be the square of the total possible number of movements the smaller square can make
(vertically or horizontally).
