EXPANDING PEG SOLITAIRE AND LEARNING ABSTRACT
ALGEBRA
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
DEPARTMENT OF MATHEMATICS EDUCATION
BRIGHAM YOUNG UNIVERSITY
The classical game of peg solitaire has a beautiful structure of underlying patterns. These patterns have a wonderful story to tell us. As we begin to play
the game we catch little glimpses of that story, like a young child poring over the
pictures in a book but not yet able to read the accompanying words.
Abstract algebra gives these patterns a language in which to tell us their story.
Once their tongue is loosed, they turn the “picture book” of peg solitaire into a rich
narrative. Algebra will provide us with three “tour guides” who not only illuminate
the game for us but, as we consider ways of either preserving or modifying these
guides, they will lead us to invent fresh, original variations on this venerably old
game. In place of an ending to the book, there is an invitation to the reader to
explore its yet-unwritten chapters.
As a child, the second author (who shall dispense with formality and use the
first person) played the game of peg solitaire on the traditional 33-hole “English
board” (see Figure 1a). I remember the tenacity of my father, filling the board
(except the center hole) with pegs, then jumping over and removing pegs until
he was stuck, and starting over again and again. In fact, he stayed up all night
until he could finally jump and reduce all the pegs down to a single one in the
center. See section 6 (game 2) for a reference on how to visualize and solve this
“Central Solitaire Game” on the English board. More recently, my parents gave
a.
b. Figure 1. The 33-hole English Board and the 37-hole French Board
me another version of the game, with decorative marbles instead of pegs, and with
1
2
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
four more holes in the board (see the “French board,” also called the “continental
board,” in Figure 1b). On the French board the Central Solitaire Game proved to
be quite challenging; my best attempts left at least two or three marbles stubbornly
remaining. The mathematician in me was intrigued, and I couldn’t rest (showing
up late for a holiday meal or two!) until I’d worked out the algebra that confirmed
my suspicion. The puzzle is unsolvable, as the reader will learn in section 5 below.
Section 8 explains a modification that makes the puzzle solvable.
The game also appealed to the teacher in me, as it is filled with tangible and
engaging illustrations of several elusive concepts in introductory abstract algebra,
such as subgroups, generators, cosets, and quotient groups. Thus, while abstract
algebra was the key to proving a version of this game to be impossible, this game
may be a key to making abstract algebra prove possible for struggling students.
I later learned that the books [1] and [3] already give a great treatment of the
underlying patterns and algebra. But a side benefit of having figured out the basic
math for myself was that new ideas began to emerge for generalizations of the game,
which do not seem to appear yet in the literature (see sections 6 through 10 herein).
The possibilities for challenging puzzles become nearly endless.
The reader should note that the books [1] and [3] go far beyond the mathematical
treatment we outline in the present paper. These excellent books cover the game
broadly, present some nice variations, teach the reader how to think about and solve
puzzles such as the standard Central Solitaire game on the English board, and go
deeper into showing whether a puzzle is solvable or not. They also give the known
history of the game and provide many references for further perusal. The present
paper’s main purposes are to outline some of our own generalizations of the game,
and to help students understand concepts of abstract algebra.
Another paper that delves into insolvability proofs is [2]. Bell explores which are
the most natural shapes of boards for peg solitaire.
For the reader who would like a quicker look at the algebra of peg solitaire, the
paper [4] gives a nice treatment with a little different point of view from ours.
Web page: The reader is also invited to visit the following web pages.
http://www.cut-the-knot.org/proofs/pegsolitaire.shtml
http://www.mathed.byu.edu/˜lawlor/Saltarri/TestPage.htm
The first page has puzzles to solve on the English board and some nice explanations.
Though not in polished form, the second page provides computer-generated puzzles along the lines of variations introduced in the present paper.
Conventions:
(1) The word layout will refer to a configuration of marbles on the board. This
includes the empty layout, with no marbles, the target layout, with just one
PEG SOLITAIRE AND ABSTRACT ALGEBRA
3
marble in the center spot of the board, and all other possibilities ranging
from one marble to a full board.
(2) All diagrams in the paper are intended to mean “use this picture or any
translation or rotation by 90, 180, or 270 degrees,” unless the diagram is of
an entire board.
(3) The word adjacent refers to a spot that is adjacent to the north, south,
east or west. The term surrounding spot includes both adjacent spots and
diagonally adjoining spots.
Note: In addition to motivating and presenting variations on solitaire, a principal
goal of the present paper is to provide material that can be read by aspiring new
students of abstract algebra.
1. Playing the Game of Marble Solitaire
Peg solitaire (hereafter called marble solitaire) in itself is an intriguing game.
The full game begins with all holes except the center filled. Alternatively, a smaller
puzzle can be posed as a setup of marbles covering only part of the board. Moves
are made by jumping a marble over an adjacent marble and landing in the vacant
spot beyond it, removing the latter marble. The usual goal is to end up with a
lone marble in the center hole of the board. Here are some puzzles to try, entitled
“Duck,” “Field goal,” and “Nevada.”
◦ ◦ ◦ ◦
◦ ◦ ◦
◦ ◦ ◦
◦ ◦ ◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
◦
◦
◦
◦
◦
◦
◦
2. How Many?
There are billions of different layouts on a 33- or 37-hole solitaire board. To
see why, start with a much simpler board, say, one having only three spots. Each
4
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
layout would then consist of zero, one, two, or three marbles on those spots, and
with a little thought the reader can find all eight possible layouts.
Now if we add a fourth spot to the board, then we get these same eight layouts
(having the new spot empty) and eight new layouts (having the new spot filled).
Add a fifth spot, a sixth spot, and so forth; with each new spot, we double the
number of layouts, ending with a grand total of 237 = 137, 438, 953, 472 different
layouts on the French board.
3. Creating solitaire puzzles
Now think about how we can create puzzles like the duck, field goal, and Nevada.
One possible approach is to begin with the target layout (just one marble in the
center) and make backwards jumps, adding in marbles instead of taking them off
(see [3], p. 817 or [1], p. xii for a quote by G. W. Leibniz on that topic). This
approach certainly guarantees that the resulting puzzle is solvable; just follow the
same jumps in reverse order. However, the method can be unwieldy if the goal is
to create a puzzle that resembles some predetermined shape.
Instead, suppose that we simply fashion a layout by placing a bunch of marbles
on the board to form a desired shape (like a swan, a sailboat, or Massachusetts).
Beasley’s book [1] (page xii) begins with a wonderful quote by Leibniz in 1710,
expressing interest in this approach to creating a puzzle layout. We may ask the
question alluded to by Leibniz: What is the likelihood that a layout formed in this
manner will be a solvable puzzle (i.e., can be reduced by legal jumps to the target
layout)? The algebra behind the game will show that the chance of this is less than
1 in 16. On the other hand, once we understand the algebra we will be able to
make a small adjustment in almost any given layout and obtain a puzzle that is
fairly likely to be solvable (see the student investigation in section 8).
4. Two algebraic tour guides: a group and a subgroup
The patterns inherent in this game can be understood by appealing to a subject
called “abstract algebra.” Abstract algebra begins with the outlandish idea that
objects other than numbers can be “added.” (Of course if you’ve worked with
vectors or matrices you won’t object to adding objects like that. But in this paper
we’ll be adding layouts to each other.) Defining this addition turns out to be a
great way of encoding and then understanding inherent patterns we find in things.
Starting with some collection of objects, we get to invent the rules for what we
want addition and subtraction to mean in our collection. The sum and difference
of any two objects must also be an object in the collection. Three commonsense
requirements are that any object minus itself equals zero, any object plus zero
equals the original object, and the associative rule A + (B + C) = (A + B) + C must
always hold true. Once we’ve managed to satisfy these three rules, our collection
PEG SOLITAIRE AND ABSTRACT ALGEBRA
5
of objects is entitled to be called a “group”, say “G.” The objects in the collection
are officially called “elements” of G. (Note: The usual definition of a group doesn’t
use the word subtraction. Instead of subtracting an element, one adds its inverse.
This formalism helps avoid algebraic errors when the group doesn’t share some of
the familiar properties of regular numbers.)
A very important thing to do with a group is to study nice subcollections from
it. A subset H of G is considered nicely behaved if when we add or subtract any
two elements of H we always get back another element of H (not just of G). Then
H deserves to be called a “subgroup” of G.
The ideas of group and subgroup are two of the three tour guides, alluded to in
the introduction of this paper, who will illuminate for us the patterns of marble
solitaire. Our third tour guide will be the idea of cosets, introduced in the next
section.
It turns out that there is a group and a subgroup that do a decent job of modelling
the game of marble solitaire. Each layout on the board is taken to be one element
of G. (If the individual marbles or spots on the board were the elements of G, then
G would only have 37 elements. Instead, G has 237 elements.)
Now we define what we mean by adding one layout to another. Suppose we
name one layout A and another one B. We define the “sum” A + B as follows. A
spot on the board is occupied by a marble in the new layout A + B if that spot is
occupied in layout A or in layout B, but not both. The spot is unoccupied in A + B
if it is unoccupied in both A and B, or if it is occupied in both A and B. So the
sum of two layouts is just their union, except that if two marbles land in the same
spot then both are removed.
Exercise: If A is any layout, what is A + A? What is A plus the empty layout?
What is A plus the full-board layout? If A + B = C, what is C + B? What is the
relationship between addition and subtraction in this particular group G?
We now notice a centrally important fact.
Proposition 4.1. (The link between solitaire and the group G.) Jumping over and
removing a marble in the game of solitaire can always represented in the algebra of
the group G by adding just the right layout having three consecutive marbles, and
having all other spots empty.
The reader should try a few jumps and see how each can be achieved by “adding”
to the given layout a new layout consisting of three consecutive marbles (see also
Figure 2 below).
Definition 4.2. We will call a layout that has all spots empty except for marbles
in three consecutive spots of some column or row a “three-bar.”
At this point an unsung hero of algebra will make a big contribution.
6
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
The humble associative law of addition is a rule that we rarely realize we’re
employing, but that has far-reaching consequences. We know that a jump can be
encoded in our algebra by the addition of some three-bar. How is a sequence of two
jumps encoded?
If we call our starting layout A, then after making one jump the new layout
will be A + D1 , for some three-bar D1 . After making a second jump, the resulting
layout will be (A + D1 ) + D2 for some three-bar D2 . Now by the associative law,
this equals A+(D1 +D2 ). In other words, the combined effect of making two jumps
is encoded in the sum of two three-bars. An example is shown below.
First jump:
◦
◦ ◦
◦ + ◦
◦
= ◦
◦
◦
◦ = ◦
◦
Second jump:
◦
◦ + Combined:

◦


◦
= 
◦
◦ ◦ ◦ ◦

◦
◦ +  + ◦


◦ ◦ ◦ ◦
◦


◦ +  + ◦
◦
so that
◦
◦ ◦
◦ + ◦ ◦ = ◦
◦
Figure 2
Similarly, the combined effect of making any number of jumps will always be
some sum of a number of three-bars. This gives us the following very powerful tool:
Suppose we have two layouts A and B, and we want to figure out whether there
exists some sequence of jumps, beginning with A, that gets us to B. (The most
important example, of course, is when B is the target layout.) In order for such
a sequence of jumps to have any chance of existing, there must be some sum of
three-bars (call the sum D) which we could add to A and get B:
A+D =B
PEG SOLITAIRE AND ABSTRACT ALGEBRA
7
B − A = D.
So given A and B, we simply subtract (or add, which is the same thing in our group
G) B and A. If the resulting layout D is a sum of three-bars then there may be a
sequence of jumps that gets us from A to B. But if D is not a sum of three-bars
then it is impossible for any sequence of jumps do accomplish the reduction from
A down to B!
Given this central observation, it becomes a question of great interest to know
how to recognize whether a layout D can or cannot be constructed as a sum of
three-bars. Here is where our subgroup H enters the stage.
Definition 4.3. We define H to be the collection of all sums of three-bars. This
subset H of G is called the “subgroup generated by the set of all three-bars.”
A few simple elements of H are pictured below (omitting surrounding spots on
the board). The point is that to get an element of H we choose any collection we
like of three-bars on the board and add them together. Then H consists of all such
sums, most of which will have many more marbles than those shown below.

◦

◦
 ◦


 ,

◦



◦

◦  ,
◦ 
◦


◦
◦


◦ ,
◦

◦




◦
◦ ◦ Now at first we may wonder whether perhaps every layout can be written as
some sum of three-bars. If that were true then H would equal G itself, in which
case H would be of no use to us whatsoever. But it turns out that only one in
sixteen elements of G is an element of H, and once we’ve studied H a little further
it will become a straightforward question to recognize whether or not a given layout
D is in H.
5. Our third tour guide: cosets
As we have said, one of the main purposes of our defining the group G and
subgroup H is to help resolve the potentially difficult question of whether a given
puzzle layout can be reduced to the target layout. The facts in the above paragraphs
represent both the good news and the bad news.
The good news is that G and H are much easier to analyze than long sequences
of actual jumps. If we have a puzzle layout A, by seeing whether A − T is in H we
take a giant leap toward knowing whether A can be reduced to the target layout T
using jumps.
The bad news is that this algebraic information, though quite a good indicator,
is not enough to fully guarantee that A can be reduced to T . We have plunged the
unsuspecting solitaire game into an algebraic environment that allows much more
than just the legal solitaire moves.
8
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
An apt comparison might be to imagine that we are trying to feel our way
through a confusing maze (the actual solitaire game), when suddenly many doors
(the algebra) temporarily appear in the walls, allowing us to move freely about and
explore the maze’s structure for a good while before we are again returned to the
start and the magic doors disappear.
Having opened up the “magic doors” then, let us spend a while exploring. Abstract algebra has a name for two layouts A and B that are related by addition
of some element of a subgroup H. If two elements A and B of a group G are
themselves not necessarily in the subgroup H but their difference D = B − A is an
element of H, we say that A and B are in the same coset of H (sometimes called
an H-coset).
In [3] these cosets are named Reiss classes for M. Reiss, who published this
algebra for solitaire in 1857 [5]; the book [1], p. 4 notes the previous discoverer
Antoine Suremain de Missery.)
Exercise: Explain why H itself is one of the sixteen cosets.
Definition 5.1. We will give a specific name to the coset of H most important to
us: Call the family that contains the target layout T (the one with just a single
marble in the center spot), the “target coset.”
Exercise: (a) Find as many layouts as you can that are in the target coset, all of
whose marbles lie within the three-by-three square at the center of the board. Hint:
Start with T and add three-bars.
(b) Let M be the total number of layouts possible in part (a), and let N be the
total number of layouts (regardless of their coset) in the center 3-by-3 square. Find
N . If you found all the answers in part (a) you will see that M divides evenly into
N . What does this suggest about how many different cosets there are?
We will see that there are exactly sixteen cosets of H, each of which contains
one sixteenth of all the billions of different layouts on the board. Thus, each coset
consists of hundreds of millions of layouts.
As an analogy, the thousands of species of mammals in the world are divided
into about twenty orders: carnivores, insect eaters, marsupials, primates, and so
forth. A basic key toward understanding two different mammals is to ask whether
they are in the same order or not, and to have some algorithm for determining the
answer. Similarly, one of the most basic questions about two solitaire layouts is
whether or not they are in the same H-coset.
In the previous section we discussed how a sequence of jumps corresponds to
adding an element of H. To recast that discussion in the language of cosets,
Any time you have a starting layout A and, by performing a sequence of solitaire
jumps, you obtain a new layout B, then A and B are guaranteed to be in the same
coset.
PEG SOLITAIRE AND ABSTRACT ALGEBRA
9
As we have pointed out, the converse of the italicized sentence above is not true;
just because two layouts are in the same coset does not guarantee that there is a
sequence of jumps to get us from one layout to the other. Rather, there are several
ways in which two layouts A and B in the same coset might be related in terms of
the legal jumps of solitaire:
(1) We might be able to get from A to B or from B to A using jumps.
(2) There might be another layout C such that it is possible to get from C to
A using jumps, and it is also possible to get from C to B using jumps. This
is likely to be true when A and B consist of relatively few marbles.
(3) There might be another layout D such that we can get from A to D with
jumps and from B to D with jumps. This is likely when each of A and B
nearly fills the board.
(4) There could be a chain of relationships like those in (2) and (3), joining
A to B. This is easiest to express if we define “reverse jumps,” exactly
backwards from jumps, in which a marble jumps over a vacant spot and we
fill that spot with a new marble. Thus item (4) is just saying that when
A and B are in the same coset, there may be some sequence of jumps and
reverse jumps that takes us from A to B.
In fact, Beasley [1], p. 63 points out that with a few trivial exceptions, any two
layouts in the same coset are related by some sequence of jumps and reverse jumps.
On the other hand, since jumps (and reverse jumps) keep us in the same coset, if
two layouts A and B are in different cosets, there will never be a sequence of jumps
and reverse jumps to get us from A to B.
Deciphering the sixteen cosets of H
To investigate these cosets our first task is to look for a small collection of rather
simple “representatives” of the cosets of H. This means that for each coset, out
of the hundreds of millions of marble layouts in that coset family, we are going to
pick just one layout to represent them all.
Definition 5.2. Let S be the square box bracketed below (chosen rather arbitrarily), and call a set of marbles on the board a “box layout” if all of the marbles lie
inside S.
Exercise: Show that for every layout X ∈ G, there is some h ∈ H such that X + h
is a box layout.
Hint: Imagine starting with some marble configuration X and beginning to
add three-bars (or combinations of them) in order
to get rid ofall the marbles
outside S. One particularly helpful combination is 1 1 1 0 + 0 1 1 1 =
1 0 0 1 . Adding this element of H has the effect of taking a marble and
10
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
leaping it over two holes (leaving the latter holes unaffected). If the leaping marble
lands atop another, both marbles are removed.
The exercise above tells us that every layout has some representative with no
marbles except within the box B. There are exactly 24 = 16 such representatives,
so there cannot be any more than 16 different cosets of H. However, in order to
conclude that there are exactly 16 cosets and no fewer (a fact that we’ve previously
alluded to but haven’t yet proved) there is still one thing to be done. We must
guarantee that two different box layouts X and Y can never be elements of the
same coset of H. Then the 16 box layouts will serve as unique representatives of all
the cosets of H, and we will have proved that there are 16 cosets. Before proceeding
to this proof, we invite interested readers to consider the following exercise that,
by naming the sixteen cosets, may make them more tangible (and fun).
Enrichment exercise: Find names for the sixteen cosets, using their box representations. Pick a category such as gem stones or trees, and find names whose vowels
reflect the marbles in the box representation for each coset. For example, if the
box representation has a marble in the first and second holes of the box (but not
the third or fourth), pick a name that contains the vowels “a” and “e” (but not
“i” or “o”). The vowels “u” and “y” are free. So for example, Ruby could be the
name for the coset whose box representation has no marbles at all, and Emerald
or Amethyst the name for the coset whose box representation has marbles only in
the first and second holes. Bismuth would be an alternative name for a coset we’ve
already named before (which one?)
Exercise: Place a bunch of marbles on the board to form some shape. Name the
layout according to the shape you’ve formed. Now find the name of the coset that
your layout belongs to, by adding three-bars or combinations of them to get rid of
marbles outside the two-by-two box.
We now return to the proof that the sixteen box layouts are all in different cosets.
We will accomplish this by defining four quantities that can be calculated for any
layout on the board, that do not change when any three-bar is added to the layout.
p
x
q y Figure 3
PEG SOLITAIRE AND ABSTRACT ALGEBRA
11
The quantities come from counting marbles in certain spots and seeing whether the
number is even or odd.
Theorem 5.3. There are exactly 16 cosets of H in G.
Proof. Consider the plaid tiling pattern below.
···
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
···
We “paint” the board in four different ways by laying this tiling pattern on the
board, shifted in different ways as indicated by the crossed circles below:
N
N
N
a: N
N
N
N
N N
N
N
p
x
N
N
p
N N
c: x
N N
N
N
N
q
y N
N
N
N N
N
N
q N
N
y
N N
N
b: N
d:
N
p
x
N N
N N
N N
N N
p
N
N
x
N N
N N
N
N
N
N
N
N
q
y
N
N
N
N
N
N
q
y
N
N N
N N
Figure 4. Painting the board to obtain invariants
Now for any layout of marbles on one of these painted boards, we count the
number of marbles that are on painted spots, and see what the parity of that
number is (0 if even, 1 if odd). The crucial fact to notice is that if we add any
three-bar to the layout and see how this affects just the painted-spot marbles, the
result will be either to add or subtract two painted-spot marbles or to leave the
number unchanged. In any case, the parity is not changed by adding a three-bar
12
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
(nor, therefore, is it changed by adding any combination of three-bars, that is, any
element of H.)
So given any layout of marbles X ∈ G let’s set that layout on each of the four
painted boards above, and define a(X), b(X), c(X) and d(X) to be the parities of
the number of painted-spot marbles in each case. We call these functions our four
“invariants,” since each function gives the same parity for all the layouts within
any given coset.
Now an examination of the box S in each of the painted boards will convince us
that for any two different box layouts X and Y , there will be some invariant that
gives a different answer when applied to X than it gives when applied to Y . Thus,
since the invariants distinguish between box layouts, they also distinguish between
entire cosets, as desired.
Exercise: What is the box representative of the target layout T ? What are a(T ),
b(T ), c(T ), and d(T )? How can you tell, then, whether a given layout is in the
same coset as T ?
Exercise: Prove that the Central Solitaire Game on the French board is unsolvable.
Exercise for students further along in an abstract algebra course: Recast the proof
of Theorem 5.6 using homomorphisms, kernels, and isomorphisms.
6. Games to illustrate the idea of cosets
(1) Representations. Students race each other or the clock to reduce given
layouts of marbles to their box layout representations.
(2) Catalyst. A “catalyst” in chemistry is an outside substance added to help
a chemical reaction occur; ideally the catalyst is left over and is recovered
after the reaction.
The idea of a catalyst marble in solitaire is explained in [3], Ch. 23
(pp. 807-809); see also [1] (p. 19). This same passage (which we highly
recommend reading) depicts a crystal-clear way of remembering the solution
to the Central Solitaire Game on the English board.
In our solitaire game of Catalyst, a layout h belonging to H is presented,
and students try to place a colored marble, called a catalyst, somewhere
on the board in such a way that a sequence of jumps will remove all the
marbles of h, leaving only the catalyst back in the same spot where the
student originally placed it. (Why is this impossible if h ∈
/ H?)
(3) Ultimate Catalyst. One of the meanings of ultimate is “the last one in
a sequence.” In the game of Ultimate Catalyst you try to be the last one
to place a valid catalyst marble on the board.
PEG SOLITAIRE AND ABSTRACT ALGEBRA
13
A layout h ∈ H with all white marbles is presented. Player 1 must
place a colored catalyst marble on the board and show how to remove the
marbles of h just as in regular Catalyst. The layout h is then restored
to the board, leaving player 1’s catalyst marble in its place on the board.
Player 2 must now place a second catalyst in a new spot and show how to
remove the marbles of h, leaving the two catalysts in their correct places.
The catalyst of player 2 must be “useful,” that is, it must jump at least
one white marble during the process of eliminating h from the board. The
original catalyst marble of player 1 may, but need not, do any of the jumps
this time around. Also, it is acceptable for the catalyst marbles to end up
“switched,” the first one ending in the beginning spot of the second and
vice versa.
The next player (either a third player or player 1 again, depending on
how many are playing) must now attempt to put a third useful catalyst on
the board. Catalysts 1 and 2 may, but need not, be used in eliminating h.
Again, the catalysts must end in the same spots as they began, possibly
having traded places.
Play continues until a player wins by placing the “ultimate catalyst,”
meaning that the following player cannot then find any place to put another
useful catalyst.
Two examples of Ultimate Catalyst
(a) If h is simply a three-bar (away from the board’s edges) then there are
four potential spots where a catalyst could be placed. (Find them.)
However, these spots are not independent, and in fact, after player 2
places the second catalyst the game ends; the other two spots will now
be blocked from placing useful catalysts there.
(b) If h consists of six marbles placed in a two by three rectangle, well away
from the board’s edges, then there are 18 potential spots for catalyst
marbles. Apparently the longest game ends after the 15th catalyst is
placed. A different sequence of choices leads to a game that ends after
only five catalysts.
(4) Acorns. This game builds on Ultimate Catalyst. As before, the beginning
setup is a layout from H, and players take turns placing useful catalysts,
which remain and accumulate on the board. We’ll call the initial marbles
“acorns,” the catalyst marbles “squirrels,” and the layout “the lawn.” The
process of jumping and removing all the acorns (subject to the same rules as
in Ultimate Catalyst) we will call “gathering the acorns.” As in Catalyst,
the gathering consists of squirrels jumping over and removing acorns, as
14
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
well as acorns jumping over and removing acorns (admittedly an unusual
thing for acorns to do).
The added option on any turn in our Acorns game is for one new
“branch” to fall onto the lawn before the next squirrel is placed. A branch
consists of three new acorns in a row (a three-bar), and must be placed
according to the following rules:
(A) After placing a new branch on the lawn, on the same turn that
player must also place a new squirrel on the lawn, and then gather the
acorns. During the gathering, the new squirrel must jump over at least one
acorn that was already on the lawn before the new oak branch fell.
(B) Each player begins the game with a limited allowance of branches
to place (say, three).
Play continues until one of the players is unable to place another squirrel.
The three lawns below are example starting positions (placed at the
center of, perhaps, a square 7 by 7 board) for Acorns. Notice that in the
third example, getting started requires a well-placed branch on the first
move; otherwise no squirrels could be placed at all.
◦
◦ ◦
◦ ◦
◦
◦
◦
◦
◦
◦
◦
◦
◦ ◦
The same game may be played as a single-player puzzle, with an initial
allotment of both branches and squirrels. The object is to successfully place
all of the given squirrels onto the board, using no more than the allotted
number of branches.
(5) Roots. Present two layouts (with only a few marbles each) that are in the
same coset, and ask students to find a third layout that links the first two;
that is, beginning at the third layout one can jump marbles and arrive at
either of the two given layouts.
(6) Sums. Present three or four layouts with several marbles each. Students
must choose two of the layouts, add them (modulo 2), and then perform
jumps until leaving a single marble in the center. This will require analyzing
the coset of each given layout and choosing which two add up to the target
coset.
(7) Master Key. This game will be presented below in the section “Changing
the coset.” We list it here because, like the above games, it gives good
practice with understanding cosets.
Student investigation: Analyze example (b) of Ultimate Catalyst, and similar
examples, to see which player can force a win, and how. The answer in each case
PEG SOLITAIRE AND ABSTRACT ALGEBRA
15
will depend on how many are playing. This investigation might include writing a
computer program.
Generalizations
We now move to a discussion of generalizations of marble solitaire. These generalizations are nicely motivated by the group-subgroup-coset structure we have set
up.
• Section 7 introduces new moves that, like the regular jump move, also
preserve the coset of a layout.
• Section 8 introduces a move that allows us to change the coset of the layout.
• Section 9 changes the subgroup H, defining a whole new set of moves for
the game.
• Section 10 changes the group G itself.
7. Shift moves
Based on our understanding of the algebra of marble solitaire, we now add an
interesting variation to the game. We can allow certain patterns of marbles to be
shifted without changing the coset or the number of marbles on the board.
Definition 7.1 (Shift moves). A shift move is characterized by the following.
(1)
(2)
(3)
(4)
One or more marbles are shifted according to a specific pattern.
No marbles are added or removed from the board.
The coset is not changed by the shift.
The destination spots for the marbles being moved must be vacant before
the shift.
There are many examples of shift moves. The simplest one, which we could call
a “leap,” consists of moving a single marble three spots away to the north, south,
east or west. Another shift with a single marble could be called a “queen’s leap,”
referring to the queen in chess. In addition to the north, south, east, and west
options, the queen’s leap also allows a marble to move diagonally three spots away.
Most of the interesting shifts involve moving two marbles. (Moving three or
more marbles can get complicated.) Shown below are the patterns for three types
of shift moves.
Exercise: Make a layout of four marbles whose positions are the two beginning
spots and the two ending spots depicted in any one of the shift moves above. What
is the coset of that layout? Why?
Exercise: In each example above, what three-bar combination can you add to the
first layout to obtain the second one?
16
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
Exercise: Find several more types of shift moves.
Shift moves with two marbles can often be classified using the moves in chess.
Notice, for example, that in the first picture above,
• each marble moves like a rook, and
• the marbles start and end a bishop move apart.
Now remove the two italicized words above and replace each of them with the
word “rook,” or “bishop” or “knight.” Each of the nine resulting sentences has at
least one valid shift move that it describes.
With these shift moves, many more puzzles in the correct coset become solvable.
(In fact, if we allowed an unlimited number of these shifts, practically every puzzle
in the correct coset would be solvable.)
Rather than allowing unlimited shifts, a typical puzzle would begin with some
layout of marbles and designate a limited “budget” of one or more shift moves of a
specific type, possibly requiring the shift to be done right at the beginning.
Example 7.2 (The town square). A puzzle we could call “The town square” begins
with the eight marbles surrounding the empty center spot of the board. The goal
is to end with a single marble in the center. This puzzle is impossible to do using
only regular jumps, since none of the eight original marbles can serve a certain
very central purpose (can you see what that is?) On the other hand, if we allow
a single shift move at the beginning, followed by regular jumps, the town square
puzzle often becomes doable. In fact, there are at least ten different types of shift
moves like the ones described above that will accomplish this purpose.
Shift moves can also be selectively introduced into other games such as the
following.
The finalist game
One well-known type of solitaire puzzle begins with a layout of white marbles
together with a single marble of a different color (say, green), which is required to
be the surviving marble in the center spot at the end. In the usual version of this
game, there are only a few spots where the green marble can start if it is to have

◦










 ↔  ◦ 

◦
◦ 


◦ 


◦   ↔  ◦ 

◦ 

 ↔  ◦
◦ 
◦




◦
PEG SOLITAIRE AND ABSTRACT ALGEBRA
17
any chance of ending in the center. However, an allowance of shift moves relaxes
this restriction and makes the green marble game more versatile and interesting.
Exercise: Solve the finalist game using the town square puzzle, with the green
marble either in the corner or on an edge. Find a solution using each of the shifts
pictured above. If the shift can be used only once and must be done at the beginning
of the jump sequence, then two of the pictured shifts work for an edge green and
the other one works for a corner green marble.
8. Changing the coset
So far an important theme has been to preserve the coset of a layout of marbles.
However, sometimes we may wish to change the coset. Puzzles that require a special
move to change to the correct coset can also be instructive for students.
One way to change coset would be to move one marble in a layout to a vacant
surrounding spot. But allowing a marble to simply be moved to a surrounding
spot during the solution of a puzzle is rather unappealing. A nice alternative is the
following, more structured move.
Definition 8.1. (The “master key” move).
Find within a layout three consecutive marbles (a three-bar).
Step 1: Move any two of the three marbles to nearby vacant spots in such a way
that the three marbles again form a three-bar, rotated 90 degrees from its previous
orientation. Note that step 1 preserves the coset of the layout.
Step 2: Now move the marble that was stationary in step 1 to any surrounding
empty spot.
Example of a master key move
◦
◦
◦
→
◦ ◦ ◦ →
◦ ◦ ◦
The term master key refers to the resemblance in step 1 to the turning of a key,
and to the fact that (depending on where the marble in step 2 begins and ends)
the master key move unlocks the passageway from any coset to any other coset.
Student investigation:
(1) Study how the master key opens the passageway from any coset to any
other. Given a starting coset and a desired ending coset, understand exactly
how to select a master key move to achieve the necessary coset change.
18
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
(2) With this skill in hand, look back at section 3 herein, and create new puzzles
for (regular) marble solitaire. Place marbles on the board to form a given
shape, then make a small adjustment to the configuration to place it in the
target coset. (This adjustment doesn’t use the actual master key move,
but only your understanding of how to adjust a coset.) See whether each
resulting puzzle is solvable.
(3) Invent and solve a collection of puzzles for the master key game described
below.
The Master Key game
A player is given a layout (a “key puzzle”) whose coset is not the target coset,
and is allowed a single use of the master key (at any point in the jump sequence)
to correct the coset and end with one marble in the center.
The first diagram below shows a key puzzle that begins with four marbles. The
square indicates the board’s center. The diagram sequence demonstrates a master
key move that corrects the coset; three regular jumps would then complete the
solution of the puzzle.
◦
◦
◦
◦
→
◦
◦
◦ ◦
→
◦
◦
◦
◦
Difficult Challenge: Solve the Central Solitaire Game on the French board
(beginning with all spots filled except the center, and ending with just one marble
in the center) using regular jumps and only one master key move. This is possible
but not easy!
Variation: As defined above, the master key move always changes the coset. Thus,
a key puzzle that allows (and requires) a single master key move must start with
a layout in one of the 15 cosets other than the target coset. To allow all 16 cosets
in this type of puzzle, one additional type of master key move could be added that
does not change the coset.
Such a move could be to start with a three-bar and shift it lengthwise one spot
without rotating it. Note that this doesn’t change the coset. Now since “the
key didn’t turn,” the doorway to a new coset is not unlocked — that is, the usual
additional marble shift (i.e., step 2 of the regular master key move) is not performed.
9. Other Subgroups of G
We have so far only investigated one particular subgroup H. A significant generalization of marble solitaire is unveiled by considering subgroups based on generators other than the three-bars. These lead to quite interesting variations on
PEG SOLITAIRE AND ABSTRACT ALGEBRA
19
the game, and can offer additional intuition to students for understanding quotient
groups of an abelian group and its subgroups.
Beasley [1], pp. 235-237 comments on the idea of using longer jumps. Instead of
jumping over a single marble, we could require a marble to jump over two adjacent
marbles and land in the empty spot beyond.
Exercise: The long jump does not preserve cosets of H the way the regular jump
does. Find a different subgroup H 0 whose cosets are preserved by the long jump.
This will mean replacing the three-bars with a different generator type.
As Beasley points out, this “twin removal solitaire” is too rigid to be very interesting, not allowing the player to develop much technique, and having relatively
few solvable puzzles. This might lead us to close the door on the idea prematurely,
causing us to miss out on a delightful wing of the solitaire edifice.
After working out the insolvability of the central game on the French board, I
was on a campout with my son, James, (then 11 years old) and was teaching him
about the mathematical patterns of the game. The creative mind of a boy wanted
to generalize the game, and my son described for me a new kind of jump you could
try. This led me to rethink the geometry of the three-bar generator, and to realize
that there were interesting substitutes for it.
Student investigation: Consider the following configurations, together with
their translations and rotations (by multiples of 90◦ ), as possible generators for
subgroups of G. (So, for example, the first subgroup to consider will consist of all
possible sums of the translations and rotations of the figure in (a).)


◦ 

(a)  ◦ 
◦ (b) ◦ ◦ ◦ ◦
!
◦ ◦
(c)
◦ ◦
(d) Multiple generator types, namely the generators from both (b) and (c)
above.
Questions:
(1) In each of the four cases above, find an upper bound for the number of
cosets for the given subgroup.
(2) Find and prove what the exact number of cosets is in each of the four cases
above. (Hint: look for invariants by painting the board in various patterns
of stripes or plaids, and checking whether generators can change the parity
of the set of marbles in the paint.)
(3) Construct a set of canonical representatives for the cosets in each case.
20
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
(4) Invent jumps, shift moves and rules for games based on these generator
types.
To get you started we suggest some possible moves based on the generators in
case (d). (In the following pictures, the ? represents the marble that is doing the
jump.)
A hurdle:
!
!
◦ ?
→
? ◦
A long jump:
? ◦ ◦
→
?
To make the new game more flexible, a nice addition is to allow, at the player’s
option, a hurdle or long jump to be preceded by one of the following coset-preserving
shifts, so long as the marbles thus shifted are then both involved in the ensuing
hurdle or long jump.
◦
◦

◦


◦
!



↔
↔
!
◦
◦


◦


 
◦
Thus, the following would be examples of legal combination moves:
!
!
!
◦ ◦
◦ ◦
→
→
◦ ◦ ◦
◦
◦
◦ !
→
◦
◦
!
◦ →
◦
!
Here are three puzzles to try using the above-described moves. The goal, as usual,
is to end with a single marble in the center. The puzzles are entitled “Mississippi,”
“Mountain,” and “Moose.”
◦ ◦
◦ ◦
◦ ◦
◦
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ PEG SOLITAIRE AND ABSTRACT ALGEBRA
21
◦ ◦
◦ ◦ ◦ ◦ ◦ ◦
◦ ◦ ◦ ◦ ◦
◦ ◦
10. Groups other than G
Modulo three
Instead of using addition modulo 2, let us now define a new group G0 similar to
G but using mod 3 addition. We will need two different colors of marbles, say, blue
and white. Each hole on the board corresponds to a matrix entry, which has the
value 0 if the hole is empty, 1 if the hole is filled with a blue marble and 2 if the
hole is filled with a white marble.
Working with the mod 3 game is important for filling in a student’s understanding, because addition and subtraction are now different operations. From the mod
2 games a student may be left with a misconception such as “two layouts are in the
same coset if their sum [rather than difference] is in the subgroup H.”
The mod 3 game also provides a fun and challenging sequel in its own right to
the regular jump game.
Student investigation:
We represent a blue marble by a “B” and a white
0
marble by a “W.” For H determined by the generators (and their rotations and
translations) in each of cases (a), (b) and (c) below,
(1) Calculate |G0 /H 0 |. (This means the number of cosets of H 0 in the group
G0 , which can also be expressed as the number of elements in the “quotient
group” G0 /H 0 .)
(2) Find several valid shift moves. Be careful to distinguish between the roles
of addition and subtraction, which were the same operation in the modulo
two case, but are now different since we are working modulo three.
(3) Invent a game for the jumps and moves you’ve found.
(4) Now try three colors of marbles, working modulo 4.
(a)
B

0

(b)  0
B
B
B
B
0
0

0

B 
0
22
ADAM CHIPMAN, GARY LAWLOR AND KEVIN POWELL
(c) Both B
B
B and
0
W
B
0
!
The Blue and White game
Here is a suggested set of rules for a one-person game using blue and white
marbles. We based the moves for this game on generator type (1) above.
Types of moves
(1) Similar to regular marble solitaire, a blue marble can jump over a blue
and a white can jump over a white, except that when the jumping marble
lands on an empty space, it changes to the opposite color. (Notice how this
corresponds to adding an all-white or all-blue three-bar.)
or
B B 0 → 0 0 W
W W 0 → 0 0 B .
(2) If two adjacent blue marbles are next to a white marble, one blue marble
can jump over the other blue and onto the white marble, eliminating both
the middle blue and the far white marble and leaving a blue marble in place
of the white. This also corresponds to adding a white three-bar. (A similar
move can be made with two adjacent white marbles next to a blue one.)
or
B B W → 0 0 B
W W B → 0 0 W .
(3) If two adjacent marbles of opposite color are next to an empty spot, they
may simply be shifted over. For example,
W B 0 → 0 W B .
(4) Two optional “squeeze” moves might be added. The first type could carry
a penalty of 1 point each time it is used, and the second a penalty of two
points per use, since it is less structured and thus easier to use than the
first.
(a) W B W → 0 W 0
or
B W B → 0 B 0 .
(b) W 0 W → 0 B 0
or
B 0 B → 0 W 0 .
Overall game
Now choose initial configurations for puzzles, making sure that the coset of an
initial configuration will allow you to end with a single marble in the center. Try
to solve the puzzles, accruing as few penalty points as possible.
One interesting puzzle consists of one blue marble in the center, surrounded by
a square of eight white marbles. This can be done with zero penalty points.
PEG SOLITAIRE AND ABSTRACT ALGEBRA
23
After gaining some experience, try this more challenging puzzle: The board is
smaller, namely a five-by-five array with just the four corner spots deleted. Begin
with blue marbles filling all but the center spot, which is left empty. End with just
a white marble in the center. This puzzle is also possible to do without any penalty
points.
11. Acknowledgments
This work was done while the third author was an undergraduate student and
the first author a recent graduate of Brigham Young University. The work was
supported by an undergraduate research grant from B.Y.U.
References
[1] John D. Beasley, The Ins & Outs of Peg Solitaire, Recreations in Mathematics series, Oxford
University Press, New York, 1985
[2] George I. Bell, A fresh look at peg solitaire, this Magazine, 80 (2007), 16-28.
[3] Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning Ways for Your Mathematical Plays, volume 4, 2nd edition, A K Peters, Massachusetts, 2004.
[4] Arie Bialostocki, An application of elementary group theory to central solitaire, The College
Mathematics Journal, Vol. 29, No. 3. (May, 1998), pp. 208-212
[5] M. Reiss, Beitrage zur Theorie der Solitär-Spiels, Crelle’s J, 54 (1857) 344-379.