Luis Martinez
Math 07 Final Paper
Professor P. Winkler
May 23rd 2013
“Gateway Puzzles- Pastimes and Gateways into Mathematics”
George Green, Carl Gauss, and Srinivasa Ramanujuan. These are just a
few of the history’s most notable self-taught and nearly self-taught
mathematicians. As a supporter of making mathematics available to the public, I
believe these men symbolize one of the greatest aspects of mathematics. This
aspect is the possibility that, with a little exposure and drive, the most unlikely
characters can contribute to the realm of mathematics. Because of this great
aspect of mathematics, I believe that mathematical puzzles that do not appear
mathematical at first, but have a huge foundation in math, are the best puzzles
for the masses. These puzzles can be attempted by anyone, with or without any
knowledge of mathematics, and serve several purposes. They are, of course,
enjoyable brain exercises: a tool to expose people who would never get involved
with anything math-related excited about math, and, most importantly, can serve
as a gateway for people never exposed to math to explore the subject. Thus,
while a mathematical puzzle may appear to be just a game, it may help get an
audience that is indifferent or that does not get much exposure to math to explore
the realm of math puzzles and math in general.
Some of the most entertaining puzzles that I have attempted in my life
appeared in magazines I found as a kid. However, as a kid, I was not fond of
math, and tried my best to avoid the subject in all forms. However, whenever I
found an old magazine, named “Ciencia Para las Mentes Jóvenes” which
translated Science for the Young Minds most of the time science magazines
aimed at a younger audience, these magazines contained puzzles. Unbeknownst
to me, these puzzles were based on simple arithmetic and geometry. Of course, I
never quite paid attention to this, and attempted the puzzles without knowing that
they were math-related. I remember one of the first puzzles involved counting up
the faces, edges, and vertices of a three-dimensional solid. Once the player
found each quantity, the player was required to put them in several boxes that
looked similar to figure 1:
The player was to put the number of vertices on the first box, the number of
edges on the second box, and the number of faces on the third, and to find the
answer. Of course, this is the most-known and simplest-known form of Euler’s
polyhedral formula. At the age of 5, I could not have known who Euler was, or
that, regardless of the shape, the answer to the problem is 2.1 The title of the
puzzle loosely translates to “Magic of Shapes Puzzle.” I kept reading the
magazine until it went out of print a couple of years later, what the magazine did
to present the puzzles was truly remarkable. It took simple math and half-decent
graphics to expose a younger audience who almost never get excited about
math. Thus, not only were puzzles like the one pictured above fun for me as a
kid, but they also helped with my indifference to math. Puzzles like these are, I
believe, by far the best kind of mathematical puzzles. They are simple to think
about, and do not appear frightening to those who have a phobia of math. This
1 Richeson, David S. 2008. Euler's Gem :The Polyhedron Formula and the Birth of Topology.
Princeton, N.J.: Princeton University Press.
layout helps those people who would never otherwise touch anything mathrelated and can help raise an interest in the subject.
The next puzzle of interest is one that goes back to the end of the 19 th
century. A popular basis for several toddler toys today, the part puzzle known as
The Tower of Hanoi has been around since Mr. Claus, Mandarin of the College
of Li-Sou-Stian, created it in 1883. A small diagram of a basic set up of the
puzzle follows in Figure 2:
The object of the game is to move the stack of disks to the pole farthest from
where the disks are initially stacked. The rules of the game are simple. A player
can only move one disk at a time, and a bigger disk cannot be stacked on top of
a smaller disk. The most common legend behind the puzzle is that somewhere in
the jungles of Southeast Asia lies a temple where there are 64 disks stacked in
the same manner as the puzzle, and, when the disks are finally stacked on the
opposite pole, the world will end2. The Towers of Hanoi are particularly
entertaining and addicting (a significant amount of my research time was spent
playing this puzzle). However, what makes this puzzle significant is its solution.
The solution to the puzzle can be found using several different methods. My
2
John Hudson, "A Brief Look at Mathematical Games and Puzzles of the 18th Century."
(unpublished working paper., University of California- San Diego, 2006),
http://math.ucsd.edu/programs/undergraduate/history_of_math_resource/history_papers/
math_history_06.pdf.
particular solution to the puzzle involves setting up a table of the fewest number
of moves to solve the puzzle. My table began with one disk, and I was able to
solve the puzzle in the fewest possible moves. I checked my solutions with the
solution posted by Emory University.3 After trying several proofs, I came about
the agreed solution for any number of n rings. The solution is given in the form:
𝑓(𝑛) = 2𝑛 − 1
The reader, if he or she wishes, can check that the solution is valid by several
methods; however, I restricted my method of checking to induction. A sketch of
the solution is attached. I find that the puzzle accomplishes something very
critical in the area of mathematics. The puzzle exposes the reader to the beauty
that is finding patterns in the most trivial parts of life like a simple game. This
accomplishment is critical, as exposing people to simple entertaining games that
do not appear mathematical can raise an interest in the actual math behind the
proof for the puzzle. Although I am not saying that arriving to the solution is easy
by any means, to quote John Dale Hudson, “[a]t it’s most basic, arithmetical and
algebraic puzzles involves simple processes that quite often appear far more
difficult then they actually are.”4 Thus, puzzles like the Towers of Hanoi are a
great way for people to venture into the simplest areas of math with something
entertaining like a stack of disks.
3
Emory University, "Towers of Hanoi." Last modified 12 09, 2002. Accessed May 15,
2013. http://www.mathcs.emory.edu/~cheung/Courses/170/Syllabus/13/hanoi.html.
4
Emory University, "Towers of Hanoi." Last modified 12 09, 2002. Accessed May 15,
2013. http://www.mathcs.emory.edu/~cheung/Courses/170/Syllabus/13/hanoi.html.
The next two puzzles of interests deal with the area of topology. One has
already been discussed at length in class, so I will not go into much detail
regarding the Seven Bridges of Königsberg puzzle.
Many high school and college students know the area of topology as the
study of Möbius Strips. The area of topology, of course, goes into much more
abstract topics than just Möbius Strips. However, the study of topology and, in
some sense, graph theory came by a similar method. Low and behold,
mathematical puzzles were at the birth of Topology. The earlier of these two
puzzles was the Seven Bridges of Königsberg puzzle. The puzzle was presented
to the famous mathematician Leonhard Euler, who refused to entertain the
puzzle at first, [The Seven Bridges of Königsberg] was one of the first few
attempts at studying the location of objects regardless of their distance.5 A
simplified sketch of the puzzle follows in Figure 4:
5 Richeson, David S. 2008. Euler's Gem :The Polyhedron Formula and the Birth of Topology.
Princeton, N.J.: Princeton University Press. 101
Legend has it that the residents of the city of Königsberg would often try to solve
the puzzle by crossing the bridges once without crossing one bridge multiple
times. While the “solution” in itself is interesting, my interest in this puzzle was
how it ended up on Euler’s desk. Clearly the puzzle had become so well-known
that somehow Euler eventually knew of its existence. A simple pastime had now
become a citywide phenomena, or at least important enough for the mayor of
Danzing Prussia, Carl Ehler, to discuss it with Euler around the time frame of
1735-1742.6 This is a great example of how puzzles that seem trivial and
unrelated to math can lead to great advancement in mathematics. I do not think
that the mayor of Danzing ever foresaw that his residents and their actions would
ever lead to the advancement of mathematics. A simple pastime at some point in
time became intriguing enough for someone think about it more than usual. As a
matter of fact, Euler himself eventually gave into curiosity and began to think
about the problem. After he published his findings, the puzzle had done a
wonderful job to raise interest in the study of the location of places and not their
distances. Thus, this puzzle had accomplished a great feat. It began as a local
pastime and led to the first studies in the area of topology.
A similar puzzle to the Seven Bridges of Königsberg, known as the Water,
Gas, and Electricity puzzle, is also a great way to get audiences interested in
puzzles in the area of Topology. The puzzle goes as follows: There are three
cottages and three utility centers: one for gas, water, and electricity respectively.
The object of the puzzle is to connect each of the cottages with all three utilities
6 Richeson, David S. 2008. Euler's Gem :The Polyhedron Formula and the Birth of Topology.
Princeton, N.J.: Princeton University Press. 101
without any of the lines connecting each utility crossing. A diagram for the
reader’s enjoyment is shown in Figure 5:
While I will not discuss the solution to this problem, I strongly suggest to the
reader that he/she attempts the puzzle to see if there exists a solution to this
problem. Similar to the Seven Bridges puzzle, the three cottages puzzle is a
great way to get large audiences more interested in topology. This puzzle
accomplishes the feat of gathering interest in the field of topology using simple
tasks and presented goals. For example, the puzzle does not seem very
mathematical in nature and as a matter of fact, I have managed to get some of
my most math-phobic friends to reach a level of near-addiction to this problem.
Clearly, the puzzle accomplishes the tasks of exposing an otherwise untapped
audience of plausible mathematicians. Once again, the area of topology
manages to appeal to a wide arrange of people and serve both as a form of
entertainment and as a great introductory tool.
The next two puzzles vary in the area of their foundation. The topics of
topology and geometry will be main focus when discussing the foundation of
each respective puzzle. Links to online simulators are provided so the reader can
attempt them.
I accidently ran across the following puzzle back when I was on a road trip
and was attempting to find directions. I realized that the colors used to color the
states on the maps were never adjacent to each other on the map. A small
diagram containing the states of Nevada, California, Arizona, Utah, Idaho, and
Oregon is shown below:7
In researching the topic, I ran across the answer to a problem I at first though of
as trivial. However, I found out that the problem goes back to the 19 th century
and has several interesting attempts at a proof. The first most prominent
mathematician to raise awareness about the problem was Francis Guthrie, who
attempted to get other mathematicians to work on it, an attempt that was futile as
most mathematicians refused to pay much attention to the problem. However.
7 Richeson, David S. 2008. Euler's Gem :The Polyhedron Formula and the Birth of Topology.
Princeton, N.J.: Princeton University Press. 136
some mathematicians, including August Möbius spent significant amounts of time
attempting to find a solution to the four-color problem. Mathematicians including
Alfred Bray Kempe made other advancements, but even as the proofs became
better, they still had shortcomings that rendered them incomplete. Even Martin
Gardner explains that he would get faulty proofs every other month on the fourcolor theorem. Eventually, a pair of computer scientists solved the problem in the
1970s with computer aid. While a pencil and paper proof has not been derived
from the complex computer proof, the mere idea of using topology to attempt the
problem is worthy of observation. Guthrie originally proposed the problem a short
time after graduating from school. Clearly, the problem was not treated as
important until it gained momentum. Even Guthrie could not gather enough
support for the puzzle and many mathematicians refused to entertain it similar to
the Seven Bridges problem. While the average person cannot solve the problem
completely, having a large audience to just attempt the problem in itself is
beneficial. The problem did a great job at this, as the problem lingered in the
math community for over a century.
The last puzzle is more modern and continues to be an entertaining
puzzle for many people today. Mathematician and puzzler Henry Dudeney
created the puzzle. His famous Haberdasher Problem is quite entertaining, and I
encourage the reader to visit the link for a simulation. The Haberdasher Problem
belongs to the branch of puzzles known as dissection puzzles. The purpose of
these puzzles is to see how many geometric shapes can be made out of other
geometric shapes and arrangements. However the mathematics behind the
Haberdahser problem can vary depending on what area the person attempting
the puzzle wants to explore. However, in my experience with the puzzle, it is
great at tapping younger audiences and getting them interested in geometry. For
example, I once attained a copy of the puzzle from the University of Mississippi
Department of Mathematics, and gave it as a gift to my younger brother. I quickly
saw him become addicted to that puzzle just trying to see why the square turned
into a triangle and vice versa. This is one of many experiences I had giving other
people puzzles that seem simple and unrelated to math. Every time, this results
in the recipients of the puzzle addicted to the puzzle and attempting to make
similar puzzles. It is important to note that these were recipients who had a small
or almost non-existent interest in math.
The puzzles mentioned in this paper are just some of what I believe to be
exemplary puzzles in the area I like to call gateway puzzles. They are made in a
way that they appeal to people who would otherwise never look at a puzzle more
than twice if they knew it was in some form or fashion mathematical. Therefore,
they serve several purposes. They appeal to the more mathematical inclined
minds that have an interest in exploring an area of math in more depth. However,
the strength of the gateway puzzles is their ability to draw in audiences who do
not like math or have ever looked at math as entertaining. They take some very
complex areas of math and make them graspable to many people; both young
minds and minds scared of math. These puzzles serve as a great form of
entertainment and a gateway for people interested and deprived of math.