Wendy Wong
July 17, 2006
Math 510
Avi Barr
Problem of the Week #3: A Marching Strip
Problem Statement:
The task is to find a formula that will determine the number of extra-strength tiles that will be needed for a
rectangular courtyard path.
The scenario is that the king wants to tile his courtyard, but the diagonal path going from one corner to the
opposite corner requires a more durable type of tile than the rest of the courtyard. This courtyard’s
dimensions may vary, but the shape will always be a rectangle. The traffic of the diagonal path of the
courtyard will be heavier than the rest of the courtyard because of visiting dignitaries; therefore, the tiles that
are along, touching, or on the diagonal line (E) must be more durable than the rest of the courtyard. All the
tiles are assumed to be square and are the same size.
C
R
E
The variables R and C are used as the number of rows and the number of columns respectively as the possible
dimensions of the courtyard. The task is to figure out a formula that determines the number of square tiles
that the diagonal line touches independent of the dimensions of the courtyard.
Process:
I approached this problem by drawing pictures, making an organized table, and finding a pattern. The first
thing I needed to do was to sketch rectangular courtyards of varying dimensions with a diagonal line going
from one corner to the opposite corner. I did this on ¼ inch graph paper, and made sketches like this:
C=7
R=8
I then shaded in the square tiles that the diagonal line goes through to find the number of “extra-strength ”
tiles that are needed for the courtyard.
C=7
R=8
I did this for many different rectangles and found inconsistent patterns. At first, I found that when you
subtract one from the sum of the number of rows and the number of columns, you will get the number of tiles
that the diagonal line crosses. When the number of rows and the number of columns are even, however, you
would subtract two from the sum of the number of rows and the number of columns. I did not find a pattern
between the area of the rectangle and the number of “extra-strength ” tiles needed. The table below includes
some of the rectangles I have found the dimensions to, the area of, the number of squares the diagonal line
crosses over, and a formula for.
Variables:
Dimensions
RxC
(Rows x Columns)
1x2
2x3
3x4
3x5
2x4
2x9
3x6
7x8
4x9
14 x 16
3x7
5x6
5x7
10 x 14
10 x 20
5x9
5 x 10
9 x 11
80 x 100
odd x even
even x odd
odd x even
odd x odd
even x even
even x odd
odd x even
odd x even
even x odd
even x even
odd x odd
odd x even
odd x odd
even x even
even x even
odd x odd
odd x even
odd x odd
even x even
Dimensions = R x C
Area = A
Number of “extra-strength” tiles = E
Greatest common factor of R and C = G
Ratio of the dimensions in simplest form = W / L
Area in
square
units
A
Number of square tiles
the diagonal line
crosses through that
must be “extrastrength”
E
Greatest
common
factor (G) of
R&C
2
4
6
7
4
10
6
14
13
28
9
10
11
22
28
13
14
19
160
1
1
1
1
2
1
3
1
1
2
1
1
1
2
10
1
5
1
20
2
6
12
15
8
18
18
56
36
224
21
30
35
140
200
45
50
99
8,000
Ratio of dimensions
in simplest form
Formula or pattern found
W/L
1/2
2/3
3/4
3/5
1/2
2/9
1/2
7/8
4/9
7/8
3/7
5/6
5/7
5/7
1/2
5/9
1/2
9/11
4/5
odd / even
even / odd
odd / even
odd / odd
odd / even
even / odd
odd / even
odd / even
even / odd
odd / even
odd / odd
odd / even
odd / odd
odd / odd
odd / even
odd / odd
odd / even
odd / odd
even / odd
G
G
G
G
G
G
G
(W+L) – 1 = E
(W+L) – 1 = E
(W+L) – 1 = E
(W+L) – 1 = E
x [( W + L ) – 1] =
(W+L) – 1 = E
x [( W + L ) – 1] =
(W+L) – 1 = E
(W+L) – 1 = E
x [( W + L ) – 1] =
(W+L) – 1 = E
(W+L) – 1 = E
(W+L) – 1 = E
x [( W + L ) – 1] =
x [( W + L ) – 1] =
(W+L) – 1 = E
x [( W + L ) – 1] =
(W+L) – 1 = E
x [( W + L ) – 1] =
E
E
E
E
E
E
E
Solution:
The formula G x [( W + L ) – 1] = E seemed to work for all the rectangles that I sketched, and when the
greatest common factor of the number of rows and the number of columns was 1, multiplying [( W + L ) – 1]
by it would not be necessary since the answer would be identical. Hence to find the number of square tiles
the diagonal line that the dignitaries walk on, you would follow the formula G x [( W + L ) – 1] = E.
Example:
Dimensions = 9 x 11
G=1
Ratio in simplest form = 9/11
W=9
L = 11
G x [( W + L ) – 1] = E
1 x [ (9 + 11) – 1 ] = E
1 x [ 20 – 1 ] = E
1 x 19 = E
E = 19
If the King ’s courtyard is 9x11 units, he needs
to purchase 19 extra-strength square tiles for the
diagonal walkway for the anticipated heavy
traffic of the dignitaries.
This formula proved to be reliable on all occasions regardless of the dimensions of the rectangle. On the
other hand, I was curious of what the formula may be if the courtyard were perfect square shape, for a square
is a special rectangle. When I sketched various squares, I found that the number of square tiles the diagonal
path crosses is the same as the length of each side of the square.
The process in which I approached the problem was appropriate though tedious. When illustrating
rectangular courtyards of various dimensions, the diagonal lines had to be drawn very precisely for me to
gather accurate counts of the number of squares the line goes through. The table I used provided more
information than I needed in finding a pattern and formula, but the formula I developed is easy to use in
determining the number of “extra-strength ” tiles needed to tile the courtyard.