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arXiv:1503.00885v1 [math.CO] 3 Mar 2015
THE BULGARIAN SOLITAIRE
AND THE MATHEMATICS AROUND IT
VESSELIN DRENSKY
Abstract. The Bulgarian solitaire is a mathematical card game played by
one person. A pack of ncards is divided into several decks (or “piles”). Each
move consists of the removing of one card from each deck and collecting the
removed cards to form a new deck. The game ends when the same position
occurs twice. It has turned out that when n=k(k+ 1)/2 is a triangular
number, the game reaches the same stable configuration with size of the piles
1,2, . . . , k. The purpose of the paper is to tell the (quite amusing) story of
the game and to discuss mathematical problems related with the Bulgarian
solitaire.
Dedicated to the memory of Borislav Bo janov (1944–2009),
a great mathematician, person, and friend.
1. The story
The popularity of the Bulgarian solitaire started around 1980. Below we present
the version of Borislav Bojanov [6] who is one of the main protagonists in the story.
The problem was brought to Bulgaria by the famous number theorist Anatolii
Karatsuba from the Steklov Mathematical Institute in Moscow. In May 1980 he
visited the Institute of Mathematics and Informatics at the Bulgarian Academy of
Sciences in Sofia. Once, after his lecture at the Seminar of Approximation Theory,
he told his Bulgarian colleagues the story of the problem.
Konstantin Oskolkov, in that time professor at the Steklov Institute, was travel-
ing from Moscow to Leningrad (now Saint Petersburg) in the night, by the fastest
train in the Soviet Union, the so called “Red Arrow”. There was another man in
his compartment and they started a conversation. When the other man learned
that Konstantin Oskolkov is a mathematician, he showed him the following game.
A pack of n= 1+ 2 + ···+kcards is divided in an arbitrary way in several packs.
Each move consists of the removing of one card from each deck and collecting the
removed cards to form a new deck. Surprisingly, it turns out that after several moves
one reaches the stable position of kpiles consisting of 1,2,...,k cards, respectively.
(The legend claims that the game was illustrated with several experiments with 15
cards.)
For example, starting with a deck of 10 cards divided in three packs of size 4, 3,
3, as in Fig. 1, we obtain a new pack of 3 cards and the number of cards in the old
packs decreases to 3, 2, 2, respectively. It is more convenient to denote only the
2010 Mathematics Subject Classification. Primary: 00A08; Secondary: 05A17, 11P81, 97A20.
Key words and phrases. Bulgarian solitaire, partitions, discrete dynamical systems, card
games.
This project was partially supported by Grant I 02/18 “Computational and Combinatorial
Methods in Algebra and Applications” of the Bulgarian National Science Fund.
1
2 VESSELIN DRENSKY
Fig. 1. A deck of 10 cards is divided in three packs of size 4, 3, 3.
size of the packs, ordering the sizes in nonincreasing order. For example, starting
from the position (4,3,3), we have marked the new size of the new pack in bold
and have consecutively
(4,3,3) ⇒(3,3,2,2) ⇒(4,2,2,1,1) ⇒(5,3,1,1)
(In the last step the two packs consisting of a single card disappear.) Then we
continue
(5,3,1,1) ⇒(4,4,2) ⇒(3,3,3,1) ⇒(4,2,2,2)
⇒(4,3,1,1,1) ⇒(5,3,2) ⇒(4,3,2,1) ⇒(4,3,2,1).
In this way we obtain the stable position (4,3,2,1).
Returning back to Moscow, Konstantin Oskolkov told the problem to the people
of the Department of Number Theory at the Steklov Institute. Anatolii Karatsuba
described this moment in the following way. “When Genadii Arkhipov (professor
in Number Theory who liked very much nice problems) learned about the problem,
his face took a Satanic expression, he ran to his office, closed the door and did not
came out until he solved the problem.”
Borislav Bojanov also liked very much nice problems. He went home, waited
until the children went to the bed and then started to think about it. Around
midnight he found a solution and was very happy. The next day he shared the
solution with some of his colleagues. Pencho Petrushev said that he also had a
solution. Milko Petkov who was an editor of the Bulgarian high school mathematical
journal “Obuchenieto po matematika” (“Education in Mathematics”) published the
problem in the section “Competition Problems” in the issue 5 of 1980. Since no
student submitted a solution, in 1981 the Editorial Board of the journal decided to
publish the solution of Borislav Bojanov [5].
Approximately in the same time the problem was published by S. Limanov and
A. L. Toom in the issue 11 of 1980 of the Russian mathematical journal “Kvant”.
The solution of Toom [38] appeared also in 1981. It contains also some analysis of
the general case of an arbitrary number of cards. It seems that [5] and [38] are the
first published solutions of the Bulgarian solitaire.
In that time the Swedish mathematician Gert Almkvist from the University of
Lund visited the Department of Algebra at the Institute of Mathematics and In-
formatics in Sofia. When he learned the problem he brought it to Sweden and told
it to his colleagues including his friend Henrik Eriksson from the Royal Institute of
Technology in Stockholm. In 1981 Eriksson wrote the paper [13] where he also pre-
sented a solution for the puzzle and gave it the name Bulgarian solitaire (Bulgarisk
patiens in Swedish). Later he visited the USA and spread the puzzle there. Jørgen
Brandt from the Aarhus University, Denmark, also learnt about the problem but
THE BULGARIAN SOLITAIRE AND THE MATHEMATICS AROUND IT 3
without its name, and in 1982 published another solution [9], where he also ana-
lyzed the general case. (Brandt starts his paper with “The problem to be discussed
in the following has been circulating for some time.”) In 1982 Donald Knuth used
the Bulgarian solitaire to start his Programming and Problem-Solving Seminar in
Stanford [22]. Finally, with the help of Ron Graham the problem reached Martin
Gardner who included it in his paper [18]. The paper by Gardner was the starting
point of the popularity of the Bulgarian solitaire among mathematicians all over
the world and was the main source of references for many years. For already 35
years the Bulgarian solitaire and its generalizations continue to inspire new research
in combinatorics, game theory, probability, computer science, and to be an object
of intensive study in research and teaching literature. Due to the efforts of Henrik
Eriksson in 2005 and the paper by Brian Hopkins [23] in 2012, recently the real
story of the Bulgarian solitaire finally reached the large audience.
2. The solution
In the first publications on the Bulgarian solitaire [5, 38, 13] the main problem is
stated in three different ways. In the Bulgarian version [5] there are k(k+1)/2 balls
grouped in mpiles. In the Russian version [38] a clerk from the Circumlocution
Office1rearranges piles of volumes of Encyclopædia Britannica. The Swedish text
[13] handles packs of cards. Nevertheless the three solutions use similar ideas. An
exposition of Tooms proof [38] with more details can be found in [21].
As we already mentioned, instead of considering packs of cards, we may consider
the sequence of the number of cards in each pack. Since we are not interested in
the order of the packs, we may order the integers in the sequence in nonincreasing
way. A finite sequence of nonnegative integers
λ= (λ1,...,λc), λ1≥...≥λc≥0, λ1+···+λc=n,
is called a partition of n. (The standard notation is λ⊢n.) The partition λ=
(λ1,...,λc) is visualized by its Young diagram [λ] (also called Ferrers diagram
when represented using dots) consisting of boxes arranged in left-justified rows,
with λiboxes in the i-th row. For example, the Young diagram of the partition
λ= (4,3,3) ⊢10 is in Fig. 2.
Fig. 2. [λ] = [4,3,3]
For our purposes it is more convenient to rotate the Young diagram on 90◦, when the
height of each row is equal to the number of cards in the corresponding pack, see Fig.
3. Then the move in the Bulgarian solitaire consists of removing the bottom row of
⇒(4,3,3) ⇒ ⇒
Fig. 3.
1The Circumlocution Office is a place of endless confusion in Little Dorrit by Charles Dickens.
4 VESSELIN DRENSKY
the (rotated) Young diagram and adding it as a column, as shown in Fig. 4. In the
⇒× × × ⇒
×
×
×⇒
Fig. 4.
language of partitions, we start with a partition λ= (λ1,...,λc) with λc>0 and
obtain the partition B(λ) = (c, λ1−1,...,λc−1). Clearly, if λi−1> c ≥λi+1 −1,
as in Fig. 5, then we assume that B(λ) = (λ1−1,...,λi−1, c, λi+1 −1,...,λc−1).
·
·
·
·
·
× × × ⇒
×
×
×
·
·
·
·
·⇒
·
·
· ×
· ×
· ×
Fig. 5.
This is a typical example of a discrete dynamical system. We consider the set P(n)
of all partitions of nand the operator B:P(n)→ P (n) which in a period of time
changes the state of the system, the partition λ, to the new state, the partition
B(λ). Hence Bplays the role of the updating function of the system. The main
problem is, starting with the initial state λ(0) to determine the state of the system
λ(t)=B(λ(t−1)) which it will reach after some interval of time t. Since we have
a finite number of states P(n) only, we may associate to the discrete dynamical
system its oriented graph with vertices the partitions λof nand oriented edges
(λ, B(λ)). In Fig. 6 we give the graph for n= 6 and n= 7.
We shall present the solution of the Bulgarian solitaire from [38] modified in
the spirit of the exposition in [9] and the solution proposed by Anders Bj¨orner,
according to the student essays [33, 19]. There are also several other solutions,
using different arguments, see, e.g., the inductive proof of Meˇstrovi´c [31].
Theorem 1. When the total number n=k(k+ 1)/2of cards is triangular, the
Bulgarian solitaire will converge into piles of size 1,2,...,k.
Proof. We use the brilliant visualization of the Bulgarian solitaire, the cradle model,
suggested by Bj¨orner. Let λ= (λ1,...,λc)⊢n,λc>0, be the partition corre-
sponding to the given collection of card packs. We turn the Young diagram [λ]
counter-clockwise by 45◦, see Fig. 7, and further consider this 45◦-turn interpreta-
tion of [λ].
Assuming that the boxes of [λ] are material points with the same mass m, we
consider the potential energy of the system
U(λ) = mg Xhij ,
where g≈9,8 m/s2is the free fall acceleration on Earth, the sum runs on all
boxes of [λ], and hij is the height of the center of the box with coordinates (i, j)
corresponding to the j-th card of the i-th pack. Clearly, hij is proportional to i+j
THE BULGARIAN SOLITAIRE AND THE MATHEMATICS AROUND IT 5
Fig. 6. The graph for the Bulgarian solitaire for n= 6 and n= 7.
⇒
Fig. 7. λ= (4,3,3).
and we may assume that it is equal to i+junits. As we discussed above, the move of
the Bulgarian solitaire removes the cboxes of the bottom row of [λ] and adds them
as the first column, as shown in Fig. 4. In this way, the box (j, 1) ∈[λ] becomes
the box (1, j)∈[B(λ)]. Obviously, in the 45◦-turn interpretation the potential
energy of these cboxes of [λ] does not change. The other n−cboxes of [λ] move
one step to the right, from the position (i, j ), j > 1, to the position (i+ 1, j −1).
Hence they also preserve their potential energy. Therefore, if c≥λ1−1, as in
Fig. 8, the move of the Bulgarian solitaire forces the boxes to cycle on the same
6 VESSELIN DRENSKY
level and preserves the potential energy of the Young diagram. If c < λ1−1, as
⇒
Fig. 8. B(4,3,3) = (3,3,2,2).
in Fig. 9, then by the gravity the excessive boxes of the second pile will fall down
southwest and the potential energy of the Young diagram will decrease. Since the
⇒ ⇒
Fig. 9. B(6,3,1) = (3,5,2) = (5,3,2).
partitions of nare a finite number, we shall reach the position when the moves do
not decrease the height of the boxes and the potential energy of the system. In this
moment, let rbe the maximal integer with the property that the first rlevels of
the Young diagram consisting of boxes (i, j ) of height i+j= 2,3,...,r+ 1, do not
have empty places. Hence these rlevels contain 1 + 2 + ···+r=r(r+ 1)/2 boxes.
If r=k, then we have reached the stable position (k, k −1,...,2,1). Otherwise,
n−r(r+ 1)/2 = k(k+ 1)/2−r(r+ 1)/2≥k > r. Hence, the (r+ 1)-th level
has an empty place and the (r+ 2)-nd level contains at least one box. Each move
pushes this box one place to the right and, when it reaches the most right position,
it starts again from the most left position. The period of the repetition of the
positions is r+ 2. Hence after several moves this box will be in position (2, r + 1).
Now we follow the position of one of the empty places in level r+ 1. It also moves
to the right with period r+ 1. Since the integers r+ 1 and r+ 2 are relatively
prime, in the moments 0,(r+ 2),2(r+ 2),...,r(r+ 2) the empty place with be
in pairwise different places. In some moment it will be in the most left position
THE BULGARIAN SOLITAIRE AND THE MATHEMATICS AROUND IT 7
(1, r + 1), as in Fig. 9. Therefore the box in position (2, r + 1) will move to the
empty position (1, r + 1), decreasing the potential energy of the system, which is a
contradiction. Hence the minimal potential energy is reached in the stable position
(k, k −1,...,2,1) only.
Now we shall consider the general case of any n. The first considerations are
in [38], the detailed study was done in [9]. Consider the oriented graph associated
with the set P(n) of all partitions of nwith vertices the partitions λ∈ P(n) and
oriented edges (λ, B(λ)). Clearly, the graph consists of several components and,
starting from any vertex of a given component, the multiple application of the
operator Bdefines a cycle which is unique for the component. In Fig. 6 the cycle
of the unique component of the graph of P(7) is
λ= (4,2,1) B
→(3,3,1) B
→(3,2,2) B
→(3,2,1,1) B
→ B4(λ) = λ= (4,2,1).
Theorem 2. Let nhave the form n= (k−1)k/2 + r,0< r ≤k, and let λ∈ P(n).
Then in the interpretation of the cradle model the solitaire will converge with a cycle
of partitions which consists of the triangular partition as bottom and rsurplus blocks
cycling above. The number of the components of the oriented graph associated with
the partitions of nis equal to the number of necklaces consisting of rblack beads
and k−rwhite beads, where the symmetry group of the necklace is the cyclic group
of order k. It is
C(n) = 1
kX
d|(r,k)
ϕ(d)k/d
r/d,
where (r, k)is the greatest common divisor of rand kand ϕ(d)is the Euler ϕ-
function, i.e., the number of positive integers ≤dand relatively prime to d.
Proof. As in the proof of Theorem 1, we shall follow the potential energy U(Bs(λ))
of the partitions Bs(λ), s= 0,1,2.... The minimum of the potential energy will
be reached when the first k−1 levels in the cradle interpretation of the diagram
[Bs(λ)] are filled in with boxes, and there are no boxes in the k+ 1-st level. We
shall identify such a partition [Bs(λ)] of minimal potential energy with the necklace
with kbeads, where the i-th bead is black if there is a box in the i-th place of the
k-th level of the diagram, and is white if the i-th place is empty, see Fig. 10. Since
the application of the operator Bmoves to the right with period kthe boxes of the
k-th level, we obtain that this corresponds to the clockwise rotation of the necklace
by 360◦/k. The number of necklaces with rblack and k−rwhite beads can be
obtained as in [9] and [1] as an easy application of the P´olya enumeration theorem,
see [4, 12].
In the case of triangular n=k(k+ 1)/2, already Toom [38] raised the problem
to determine the longest path in the graph of P(n)to reach the stable partition
σ= (k, k −1,...,2,1). He showed that, starting from the partition τ= (k−1, k −
1, k −2, k −3, k −4,...,3,2,1,1), the minimal swith the property σ=Bs(τ)is
s=k(k−1). Knuth [22] checked the equality
σ= (k, k −1,...,2,1) = Bk(k−1) (λ), λ ∈ P(k(k+ 1)/2),
for k≤5. He asked his students to write a computer program to check it for k≤10
and conjectured that this holds for any k. The conjecture of Knuth was proved by
Igusa [28] and Bentz [3]. Bentz also established the following interesting property
8 VESSELIN DRENSKY
⇒ ⇒ ⇒
m
Fig. 10. The cycle generated by (5,3,3,1) and the corresponding necklace.
of the partition τconsidered by Toom: The partitions Bi(τ)and Bk(k−1)−i−1(τ)
are conjugate for i= 0,1,2,...,k(k−1) −1.This means that the related Young
diagrams are obtained by reflection with respect to the bisectrix from the origin of
the first quadrant of the coordinate plane, i.e., the lengths of the columns of one
diagram are equal to the lengths of the rows of the other. The general case of an
arbitrary nwas studied by Etienne [16].
In the theory of cellular automata, a Garden of Eden configuration is a config-
uration that cannot appear on the lattice after one time step, no matter what the
initial configuration. In other words, these are the configurations with no predeces-
sors. The terminology comes from the foundational paper [32] by analogy with the
concept of the Garden of Eden which, following Semitic religions, was created out
of nowhere. Hopkins and Jones [25] studied the Garden of Eden partitions (GE-
partitions) defined by the property that they do not belong to the image B(P(n)).
It has turned out that each cycle in the oriented graph of P(n)can be reached from
a GE-partition. We shall mention only the following easy property and refer to
[25, 26, 24, 27] for more details and further developments.
Proposition 3. A partition λ= (λ1,...,λs)⊢n,λs>0, is a GE-partition if and
only if λ1< s −1.
3. Generalizations
Before the paper by Hopkins [23], only pieces of the history of the Bulgarian
solitaire were known by the large mathematical community. A couple of times the
solitaire was rediscovered or called with other names. The case for triangular nis
known also as the Karatsuba solitaire. (Since Karatsuba brought it to Bulgaria some
THE BULGARIAN SOLITAIRE AND THE MATHEMATICS AROUND IT 9
people claimed that he invented the puzzle.) We shall discuss several generalizations
of the Bulgarian solitaire which have been studied in the literature.
3.1. Real life interpretation of the Bulgarian solitaire. Discrete dynamical
systems often have economic or biological interpretations. The Bulgarian solitaire
reflects the following situation from the real life. Consider a company consisting of a
number of departments. The Board of Directors decides to create a new department,
but does not want to increase the total number of employees. So, the Board takes a
member from the existing departments and move the person to the new department.
If we assume that the number of cards in the piles is equal to the number of
persons in the departments, the Bulgarian solitaire corresponds to the “greediest”
case, when the new department is formed by taking a person from each department
of the company.
3.2. Austrian solitaire. Inspired by a discussion on the so-called Austrian school
of capital theory, Akin and Davis [1] introduced the Austrian solitaire which has
the following economic interpretation. A company has several machines. Each
machine has, when new, a life of exactly Lyears. Each year for each machine on
line the company deposits 1/L of its cost into the bank as a sinking fund. Then
it buys as many new machines as it can afford, and the remaining funds are left
in the bank until next year. Now, take a pack of cards and divide it in piles in
such a way that each pack contains not more than Lcards. Think of the piles as
machines. The number of the cards is equal to the number of productive years left
for a particular machine. One of the piles is specific. It is the bank and does not
correspond to a machine. Each move of the solitaire consists of two steps. In the
first step we remove one card from each of the ordinary piles (the machines have
one year less to live) and add the cards to the pile of the bank. In the second step we
take Lcards from the bank and form a new ordinary pile of size L(we buy a new
machine) and continue this process until the bank contains < L cards. The problem
is to describe the cycles of the corresponding dynamical system. For enumeration
problems related with the Austrian solitaire see the Baccalaureate Degree Thesis
of Bastola [2].
3.3. Carolina solitaire. When visiting the University of South Carolina, Columbia,
Andrey Andreev from the Institute of Mathematics and Informatics at the Bul-
garian Academy of Sciences introduced a new ordered variation of the Bulgarian
solitaire called the Carolina solitaire. The game begins with ncards divided into a
row of piles of sizes α1,...,αc,α1+···+αc=n,αi>0.Hence we work with com-
positions, i.e., ordered systems of positive integers, (α1,...,αc) rather than with
partitions λ= (λ1,...,λc)⊢n.The move consists of removing one card from each
pile, and then placing these ccards in a pile ahead of the others. Any exhausted
pile (of size 0) is ignored and only nonempty piles are considered. In other words,
if Cis the operator of the Carolina solitaire, then, up to the ignored empty piles
(obtained for αi= 1), the action is defined by
C(α1,...,αc) = (c, α1−1,...,αc−1), αi>0.
For a triangular number n=k(k+ 1)/2this new game also appears to arrive at a
stable division, with piles of sizes k, k −1,...,2,1.Griggs and Ho [20] derived upper
and lower bounds for the maximum number of moves required to reach a cycle of
10 VESSELIN DRENSKY
the graph of the compositions of n.See also the Master Thesis of Tambellini [37]
for other properties of the Carolina solitaire.
3.4. Montreal solitaire. It is suggested by Cannings and Haigh [11]. The po-
sitions are compositions α= (α1,...,αc) of nonnegative integers. Identifying the
compositions (α1,...,αc), (0, α1,...,αc), and (α1,...,αc,0), we may consider only
the case when α1and αcare positive. The successor rule Mof the Montreal soli-
taire is defined in the following way. If all αiin α= (α1,...,αc) are positive,
then
M(α) = (α1−1,...,αc−1, c).
Then we extend the action of Minductively. If
α= (β, 0,...,0
|{z }
rtimes
, γ) = (β , 0r, γ), β = (β1,...,βc), βi>0, γ = (γ1,...,γd),
then
M(α) = (M(β, 0r−1,M(γ)),
keeping the 0’s in the beginning of M(γ). For example,
M(1,0,2) = (0,1,1,1) = (0,1,1,1)(= (1,1,1))
and
M(1,2,0,1,0,2) = (1,2,0,1,1,1).
Another example is
(3,2,2) M
→(2,1,1,3) M
→(1,0,0,2,4) M
→(1,0,1,3,2) M
→(1,0,2,1,3) M
→(1,1,0,2,3)
and one can check that M18(3,2,2) = (3,2,2). In contrast to the Bulgarian soli-
taire, in the Montreal solitaire each position αhas a unique predecessor M−1(α)
and there exists a positive integer psuch that Mp(α) = α. Hence the set of compo-
sitions (α1,...,αc)of nwith positive α1and αcis a union of disjoint cycles. We
refer to [11] for more properties of the game.
3.5. Other discrete generalizations. There are also several other games moti-
vated by the Bulgarian solitaire. As in the case of the regular Bulgarian solitaire,
the problems studied concern the type and the number of cycles, the Garden of
Eden positions, etc. We shall list a couple of generalizations.
Locke [30] invented the Red-green Bulgarian solitaire where the cards are colored
in two colors, red and green, and the moves depend on the existence of green cards
in each pile.
Grensj¨o [19] studied the Three-dimensional Bulgarian solitaire. The idea is to
define the game on plane partitions, which can be visualized using three-dimensional
Young diagrams.
¨
Ohman [33] considered two generalizations: the Dual Bulgarian solitaire and
the Multiplayer Bulgarian solitaire. In the dual game the piles are ordered in
nonincreasing order. In each move the largest pile is removed and its cards are
distributed to the remaining piles one by one, from larger to smaller, with any ex-
cessive blocks forming piles of size 1. For example, the partition (4,4,3,2,2,1,1)
goes to (5,4,3,3,1,1) = (4 + 1,3 + 1,2 + 1,2 + 1,1 + 0,1 + 0), and (6,6,3,2,1) goes
to (7,4,3,2,1,1) = (6 + 1,3 + 1,2 + 1,1 + 1,0 + 1,0 + 1). In the regular Bulgarian
solitaire we may assume that each part of a partition corresponds to the income of
a citizen or a company. Then the Government collects the same taxes from each
THE BULGARIAN SOLITAIRE AND THE MATHEMATICS AROUND IT 11
person and each company and uses the collected money to make a new company.
In the dual game, see the comments in [19], one applies the principles of Robin
Hood: taking from the rich and giving to the poor. It can be shown that if Robin
Hood continues to take from the rich and give to the poor, then the distribution of
fortune in his community will become close to triangular. The explanation is sim-
ple. If one translates the game in the cradle model, it is easy to see that the dual
Bulgarian solitaire is equivalent to the regular one. Bouchet [7, 8], see also Bruhn
[10], established that ⁀
the dual Bulgarian solitaire corresponds to the old African
game Owari which consists of cyclically ordered pits that are filled with pebbles. In
a sowing move all the pebbles are taken out of one pit and distributed one by one
in subsequent pits. Repeated sowing will give rise to recurrent states of the owari.
One can interpret the multiplayer game in the following way. Several players
sitting around a circular table play the Bulgarian solitaire. All players remove one
card from each of their piles at the same time and then pass this new pile to the
player on their right. In other words, if we have a collection of partitions
λ(1) = (λ(1)
1,...,λ(1)
c1),...,λ(s)= (λ(s)
1,...,λ(s)
cs), λ(i)
ci>0,
the move sends λ(i)to (ci−1, λ(i)
1−1,...,λ(i)
ci−1), where by convention c0=cnand
the parts of the image of λ(i)are rearranged in nonincreasing order if necessary.
Servedio and Yeh [36] suggested a game which can be interpreted in the following
way. There are cplayers sitting around a circular table. The i-th player has αi
cards. (We consider circular compositions on n, identifying α= (α1,...,αc) and
(αc, α1,...,αc−1).) The move consists of the following simultaneous actions of the
players. The i-th one takes one’s cards and distributes them clockwise, to oneself
and to the following αi−1players.
Janetzko in his Ph. D. Thesis [29] considered a similar game with cplayers
around a circular table and with total number of ncards. A pointer points one
of the persons (e.g., the i-th one) who takes all cards from his or her pile and
distributes them to all players on the right, giving one card to the (i+ 1)-th player,
one card to the (i+ 2)-th player, etc. (the addition is modulo c). At the end
the pointer points at the player that receives the last card. Repeating this procedure
gives a periodic sequence of pointer positions. The thesis studies the question which
periodic sequences can be realized as such pointer sequences. It is interesting to
mention that the problem is reduced to the investigation of an inhomogeneous linear
system of equations. Then the author applies the Perron-Frobenius theorem, [34]
and [17], which asserts that a real square matrix with positive entries has a unique
largest real eigenvalue and that the corresponding eigenvector has strictly positive
components, and also asserts a similar statement for certain classes of nonnegative
matrices.
3.6. Stochastic Bulgarian solitaires. There are many possible ways to formu-
late stochastic versions of the Bulgarian solitaire. Popov [35] introduced his Ran-
dom Bulgarian solitaire. As in the regular Bulgarian solitaire, a deck of ncards is
divided into several piles. Then one fixes a number p∈(0,1] and, for each pile,
one leaves it intact with probability 1 −pand removes one card from the pile with
probability p, independently of the other piles. The cards that are removed are
collected to form a new pile. For p= 1 this is the regular, or deterministic, Bul-
garian solitaire. The model with parameter 0< p < 1is a discrete-time irreducible
12 VESSELIN DRENSKY
and aperiodic Markov chain on the space of unordered partitions of n. For the sta-
tionary measure of the game Popov proves that most of its mass is concentrated on
(roughly)triangular configurations of a certain type. Eriksson and Sj¨ostrand [15]
showed that the random Bulgarian solitaire can be interpreted as a birth-and-death
process on Young diagrams.
Recently Eriksson, Jonsson, and Sj¨ostrand [14] introduced another Stochastic
Bulgarian solitaire. They assume that the selection acts on the cards rather than
on the piles: When forming a new pile by picking cards from the old piles, every card
is picked with a fixed probability 0< p < 1, independently of all other cards. They
establish a surprising fact. The solitaire is not drawn to triangular configurations
but to an exponential shape.
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Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad.
G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
E-mail address:drensky@math.bas.bg
