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Date
TABLE OF CONTENTS
ABSTRACT…………………………………………………………………1
INTRODUCTION…………………………………………………………...2
THE PROBLEM……………………………………………………………..3
QUESTION ONE……………………………………………………………4
QUESTION TWO…………………………………………………………..14
QUESTION THREE………………………………………………………..16
QUESTION FOUR…………………………………………………………17
CONCLUSION……………………………………………………………..18
REFERENCES……………………………………………………………..19
Abstract
This problem is based on the game of pool played on a square pool table. It is
theoretical in that the ball is not affected by external influences such as spin and
velocity. When it hits a wall, it bounces off that wall and continues along its path
until it falls into a pocket. There are four pockets on the table, one in each corner.
The pockets, along with the ball, have no measurable radius. This study explores
aiming the ball at different points along the top of the table and determining the
number of bounces it will make before it falls into a pocket and which pocket it
will fall into. Then, the radius of the pocket is increased to keep the ball from
traveling the full theoretical distance and those questions are revisited. Finally,
the situation is explored using irrational numbers.
-1-
Introduction
The game of pool is an active and engaging pastime historically enjoyed by
royalty, nobles, crooks and countrymen. A pool table, some good friends and
refreshments are all the makings of a fun Saturday night. But have you ever
considered the game of pool from a mathematical point of view? It‟s a plethora
of geometry, algebra, number sense and logic all rolled together and disguised as
a game.
The game of billiards has been around since the 1600‟s. Shakespeare even
mentioned it in „Anthony and Cleopatra‟. The table itself has undergone many
changes but by 1850 the billiards table had evolved to its current form.
Traditionally called “billiards”, the game has taken on the name of “pool”. The
word pool means a collective bet or ante and, although many games involve a
wager, the word “pool” has stuck to the game of billiards specifically.
Up until recently, the game has been dominated by men. In the 1920‟s, men
gathered in pool halls to loiter, smoke, fight, bet and play. However, women have
enthusiastically played pool throughout history as well and are now making
strides in becoming just as involved with the game as men.
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The Problem
We play pool on a square table that is 1 unit by 1 unit. Our game is in that
the pool ball, when struck, will keep moving, bouncing off the sides, until it
falls into a pocket. Also our pool ball is small enough that we can think of it
as having no measurable radius. There are pockets in all 4 corners and
nowhere else. Each pocket is a quarter circle centered at the corner. The
size of the pocket is measured by it’s radius in pool table units, and we will
always assume that all pockets are the same size. A ball that hits any part of
a pocket will fall into a pocket. We always start with the ball in the bottom
left corner when we strike it.
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Question One
Suppose we aim at a point on the top horizontal side that is 13/19 units from
the left. Suppose the pocket is also small enough that the radius has no
measurable radius. Thus our small pool ball must hit a corner exactly to fall
into the pocket? Will the ball eventually fall into a pocket? If so, what
pocket does it fall in, and how many bounces does it make before it falls?
This problem has multiple entry points and, to solve it, one must decide where
they are most comfortable beginning. All learners, children and adult, possess
various skills in different areas of mathematics and utilize those skill sets to
develop different perspectives on problems. One who has a very strong
visual/spatial sense may see this as strictly a geometry problem and would
approach it by thinking about the angles that the ball travels within the square.
There may be the prior knowledge that, in the game of pool, the angle of
reflection equals the angle of incidence when the ball strikes a point on the side of
the table. However, the task of drawing a square pool table of 247 square units,
(the least common multiple of 13 and 19 is 247), is nearly insurmountable.
Or, maybe one is more inclined towards algebra and would begin by determining
the slope of the line the ball travels in order to solve this problem. One could
begin the problem by looking at the size that the square pool table would need to
be. In the case of 13/19, the table would need to have an area of 247 square units
because the least common multiple of 13 and 19 is 247. Yet again, the task of
drawing a table that large complicates matters too much to continue with.
-4-
So here we have it…two different perspectives on how to approach this problem
using basically the same steps and running into the same obstacle. The next
logical step would be to solve a simpler problem with the hope that the results
could be expanded and a solution thusly found. This, once again, brings up
multiple entry points and perspectives. Is there another way to look at the aim
point? Maybe there is some flexibility in this aspect of the problem that could
help us better understand it. Maybe, instead of aiming at the top of the square, we
could aim at the right side of the square. Is that possible? Would it derive the
same results? If we take the original table and flip it across the y-axis and then
rotate it 90 degrees clockwise, the table takes on the appearance it would if we
had aimed at a point halfway up the right vertical side. (See figures 1 and 2)
In figure 1, we aim at a point that is ½ the
distance from
the top left side.
figure 1 – aiming1/2 way across the top from the left
Figure 2 shows figure 1 after a flip across
the y-axis and a 90 º clockwise rotation.
figure 2-figure 1 after a flip and rotation
-5-
Of course, before we can proceed, we must consider how this will affect the slope
of the line.
In figure one, when starting from the lower left corner and aiming at a point ½ the
distance from the top left, the fraction provides the slope of the line in the
traditional form of rise/run. So, for every 1 unit up, we must travel ½ unit over.
On a 4 x 4 square, the vertices that the ball will travel are as follows: (1/2, 1), (1,
2), (1 ½, 3), (2, 4). However, (2, 4) is a bounce against the top at which point the
ball will travel in the opposite direction, (the slope is multiplied by -1), and hit the
following vertices: (2 ½, 3), (3, 2), (3 ½, 1), falling into the lower right pocket.
However, in figure two, when starting from the lower left corner and aiming at a
point ½ the distance up from the bottom right, the slope of the line is the inverse
of the original slope to compensate for the flip and rotation that we performed
earlier. So, for every ½ unit up, we travel one unit over. On a 4 x 4 square, the
vertices that the ball will travel are as follows: (1, ½), (2, 1), (3, 1 ½), (4, 2).
Again, (4, 2) is a bounce against the right vertical side at which point the slope is
multiplied by -1 and the ball will travel in the opposite direction, hit the following
vertices: (3, 2 ½), (2, 3), (1, 3 ½), falling into the upper left pocket. The pocket
change is also due to the flip across the y-axis and the 90º clockwise rotation.
Let‟s pause here and consider what we already know:
-Neither the ball nor the pockets have any measurable radius. The ball is
basically a point that is traveling along a graph. Since the pockets have no
measurable radius, it is critical that the ball hit a point exactly to fall into a pocket.
Even a slight variance will cause the ball to continue the bouncing pattern.
-6-
-The angle of reflection equals the angle of incidence. Therefore, the ball will
bounce off of any side at the same angle at which it hit that side.
-It does not matter if we look at aiming at the top horizontal or the right vertical
side as long as we compensate for the change in slope.
-The pool table is a square so to draw out a physical model we will need to use the
least common multiple of the two dimensions in order to get an accurate
representation of the table.
-We know that if we aim, (from the lower left corner), at a point that is half way
across the top, the ball will bounce once and land in the lower right pocket.
-Whenever the ball hits a wall, the slope is multiplied by -1 and the ball changes
direction.
At this point we can look for a simpler problem to see if there are any patterns
that can direct us towards the answer to the original question. Start small. What
happens when we aim at a point 1/3 up the right vertical side? 2/3? 1/4? As data
is gathered, it is imperative to keep it organized in such a way that we can see any
emerging patterns and extend those patterns.
Inevitably, after the simpler problems are solved, we are still faced with the
original, large and difficult problem so we need to find a way to work more
efficiently. Sometimes the extended patterns still do not tell the entire story.
What if we distort the square pool table into a rectangle with the dimensions of
the fraction we are aiming for? Although the table will not be technically
accurate, the slope will be accurate even though it will look like a one-to-one
ratio.
-7-
As long as we keep in mind that the visual representation is distorted, we can use
rectangular tables to discover how many bounces will occur and what pocket the
ball will land in. It is a good idea to test any theory with simpler numbers to
ensure that it is correct before expanding it to the less manageable fractions.
An interesting pattern emerges among the unit fractions, represented in figure 3,
below.
Fraction
Number of bounces
Pocket that the ball
lands in
1/2
1
Upper left
1/3
2
Upper right
1/4
3
Upper left
1/5
4
Upper right
figure 3-patterns created by aiming at unit fractions
For each unit fraction the number of bounces is one less than the denominator.
Also, whenever the denominator is even, the ball falls into the upper left pocket
and whenever it is an odd number, the ball falls into the upper right pocket.
Interestingly, it seems that the ball never falls into the lower left pocket from
which it started, leaving only three pockets as possible landing points. Because
the ball travels across the square table, bouncing off the walls at reflected angles,
it is physically impossible for the ball to ever land in the corner it started from. If
we exclude the pocket that the ball actually falls into, and the starting pocket,
there are two un-used pockets that we will have to subtract when we develop a
formula for finding the number of bounces.
-
8–
Looking back at the distorted tables for the unit fractions, we notice that the ball
traversed every single square in the graph paper in order to reach a pocket, (see
figure 4).
figure 4-the path the ball travels through unit fraction distorted pool tables
What would happen if we worked with a fraction that could be reduced? Would
the number of bounces be the same? Would the ball cross through every single
square in the table? See figure 5, below.
-9-
figure 5-the path of the ball through distorted square pool tables
When the fraction is not reduced, the ball still bounces the same number of times
it does when it is reduced but more squares get crossed by the ball on the
diagonal. However, if we divide the number of traversed squares by the greatest
common factor of the two dimensions, we get the same number of crossed squares
as we did in the reduced fraction.
- 10 –
Example: The path of the ball when aiming at a point 2/10 up the right side
traverses ten of the squares in the table. 10 ÷ 2 (the greatest common factor of 2
and 10) = 5. 5 is the number of squares crossed by the ball in the 1/5 table.
In fact, when looking at 2/10, the ball traverses ten of the diagonals. And in 2/8,
it traverses eight of the diagonals. In 1/3, the ball goes through three of the
squares diagonally, in 1/4 it goes through four of them and in 1/5 it goes through
five of them. The ball diagonally crosses the number of squares equal to the least
common multiple of the two dimensions of the distorted pool table.
Let‟s pause here and add to our knowledge bank the information that we‟ve
recently acquired.
-The ball can really only fall into one of three possible pockets, leaving two of
them out of the equation completely.
-The greatest common factor plays a pivotal role in determining the number of
bounces the ball will make.
-The least common multiple is used to determine how many squares inside the
pool table will be crossed diagonally as the ball travels to the pocket.
Now it‟s time to step back and see if we can come up with a formula to help us
determine the number of bounces for any aim point. See figure 6, below.
- 11 -
figure 6-aiming the ball at unit fractions, gcf of the dimensions and number of
bounces the ball will make before landing in a pocket.
Fraction
GCF
Number of bounces
1/2
1
1
1/3
1
2
1/4
1
3
1/5
1
4
2/8
2
3
2/10
2
4
Using 2/8 as an example, if we add the two dimensions together, 2 + 8 = 10, then
divide by the greatest common factor, 10 ÷ 2 = 5, we have a number that is two
more than the correct number of bounces. We need to subtract the two un-used
pockets to arrive at the number of bounces the ball will make.
[(l + w) / gcf] – 2 = number of bounces.
Using that formula to figure out how many bounces the ball would make in the
original problem of aiming at a point 13/19 from the top left before it falls into a
pocket, we just plug in the numbers as follows:
[(13 + 19)/1] – 2 = number of bounces
[32 ÷ 1] – 2 = number of bounces
32 – 2 = 30 bounces.
- 12 -
If we aim at a point that is 13/19 from the top left side, the ball will bounce 30
times and will fall into the upper right pocket.
- 13 -
Question Two:
Suppose we aim at a point on the top horizontal side that is 13/19 units from
the left. What is the smallest size we can make our pockets that will cause
the ball not to go the full theoretical distance in the last problem. If we use a
pocket of that size, what pocket does the ball fall in, and how many bounces
does it make before it falls?
It seems that the largest dimension, written as a unit fraction would be the
smallest possible radius a pocket could have that would keep the ball from
traveling the full theoretical distance. Therefore, my conjecture for the smallest
pocket radius when aiming at a point 13/19 from the left on the top side is 1/19
units. See figure 7, below.
figure 7-aiming at a point 13/19 from the top left side
- 14 -
As the table shows, if the upper left pocket had a radius of 1/13 units, the ball
would have a much shorter trip and would have landed in that pocket. But, with a
smaller radius of 1/19 units, the ball bounces one more time, for a total of four
bounces, and falls into the upper left pocket, making a significantly shorter trip
than was made with the zero radius pockets.
Essentially, the ball travels in a straight line along the slope determined by the
aim point until it hits a wall. At that point, the slope is multiplied by -1 and the
ball changes directions. When it hits the next wall, the slope is again multiplied
by -1 bringing it back to the original slope. Figures 8 and 9 illustrate what occurs
if we “unfold” the bounces. The diagram shows that the ball would travel in a
straight line until it falls into a pocket.
figure 8-bounces bounded
figure 9-“unfolded” bounces
by the pool table edges
- 15 -
Question Three:
Suppose we aim at a point on the top horizontal side that is 87/99 units from
the left. Can you answer the questions above?
Using the formula [(l + w)/gcf] -2, we can figure out how many bounces the ball
will make before it hits a pocket.
[(87 + 99)/3] – 2 = number of bounces
[186/3] – 2 = number of bounces
62-2 = 60
The ball will bounce 60 times and will land in the upper right pocket.
When the pocket size is increased to a radius of 1/99 units the ball will bounce 14
times and fall into the lower right pocket.
In figure 10 we can see the path that the ball will travel on the distorted square
pool table. Because 29/33 is proportionately the same as 87/99, the pocket size of
1/33 in the diagram is equal to a pocket size of 1/99 on a square table. Therefore,
we see that the ball will bounce 14 times and land in the lower right pocket.
- 16 -
figure 10-the path of the ball when aimed at a point 13/19 from the top left.
-17-
Question Four:
Suppose we aim at a point on the top horizontal side that is 1/ √2 units from
the left. Can you answer the questions above?
No, the above questions would be impossible to answer if we aim at a point on the
top horizontal side that is 1/ √2 units from the left because √2 is an irrational
number and, due to that, the quotient of 1/ √2 is also irrational. Therefore, any
pool table that we could draw as a model would be an estimate, at best, because
we would have to round the irrational number at some point.
How can we prove that the square root of two is irrational?
Suppose the square root of two is a rational number that has been reduced to
lowest terms.
In that case,
√2 = n † m. Since it has been reduced, either n or m must be odd, the other even.
2 = n^2 ÷ m ^2
2m^2 = n ^2
2m^2 = (2k)^2 We use 2k as a substitute for n because we are assuming that n is
positive and two multiplied by any positive number yields a positive number.
(2m^2) ÷2 = (4k^2) ÷2
m ^2 = k (2k)
- 18 -
2k is even because anything multiplied by an even number equals an even
number. Because m ^2 = k (2k), m^2 is also even. Two even numbers cannot
form a reduced fraction, therefore, √2 is irrational. We have proof by
contradiction.
Further, in looking at √2 as a continued fraction we have the following:
√2 = 1.414213562 = 1 + .4142356
Now we divide the decimal part of the quotient by one as follows:
1 ÷ .41421356 = 2.41421356
This equals 2 + .41421356. If we continue to divide the decimal part of the
quotient by one, we end up with a repeating loop of 2.41421356.
We have returned to the original decimal approximation that we started with so
the continued fraction for √2 = [1; 2, 2, 2, 2,]. While we can get closer and closer
to a rational number, we never actually reach it. Therefore, the slope of
1/ √2
will never hit a pocket; a ball aimed at that point will just bounce around the table
infinitely.
- 19 -
Conclusion:
Mathematician W.S. Anglin once said, “Mathematics is not a careful march down
a well-cleared highway, but a journey into a strange wilderness, where the
explorers often get lost. Rigour should be a signal to the historian that the maps
have been made, and the real explorers have gone elsewhere”.
This billiards problem is rich with the applied mathematical concepts of slope,
greatest common factor, least common multiple, patterns and functions along with
angles. It weaves the strands of algebra, number sense and geometry into a real
life situation that is accessible to a large audience. It is an example of the
fascinating beauty of mathematics which lies below the surface of common
situations. True mathematics problems do not have one algorithmic solution path
that stems from applying a formula; they offer multiple entry points and many
solution paths offering triumphs and tribulations along the journey to the solution.
- 20 -
References:
Anglin, W.S. “Mathematics and History”, Mathematical Intelligencer, v.4, no. 4
Billiards Game Applet, www.ies.co.jp
Brown, Robert B. “Pool Table Geometry”, www.ohiorc.org, The Ohio State
University
Billiards, www.mathworld.com/Billiards.html
Douglas, Dr. “A Slope Problem Involving a Billiards Table”,
www.mathforum.org/dr.douglas, Oct. 4, 2004
www.mymathforum.com
Shamos, Mike. “A Brief History of the Noble Game of Billiards”,
www.billiardsdigest.com
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