AbstractOwari is an old African game that 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. Bouchet studied such periodical states in an idealised setup, where there are infinitely many pits. We characterise periodical states in owaris with finitely many pits. Our result implies Bouchet's result

Bruhn, Henning

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Periodical states and marching groups in a closed owari 

Discrete Mathematics 308 (2008) 3694–3698
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Periodical states and marching groups in a closed owari
Henning Bruhn
Mathematisches Seminar, Universität Hamburg, BundesstraYe 55, 20146 Hamburg, Germany
Received 7 February 2006; received in revised form 25 June 2007; accepted 5 July 2007
Available online 4 September 2007
Abstract
Owari is an old African game that 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.
Bouchet studied such periodical states in an idealised setup, where there are inﬁnitely many pits. We characterise periodical states
in owaris with ﬁnitely many pits. Our result implies Bouchet’s result.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Mathematical games; Periodical states
1. Introduction
Owari is an over 1000 years old game of African origin that is now played worldwide. It and its variants are known
under many names, some of which are oware, awalé, warri and awari. Its gameboard consists of a number of cyclically
arranged pits or “houses”, typically 12 of them, each of which is ﬁlled with 4 seeds or pebbles at the beginning of the
game. In a turn, one of the two players takes all pebbles out of one pit and distributes them one by one in subsequent
pits in counterclockwise direction. This is called sowing. After having sown the player captures certain pebbles. The
player who captures more than half of all pebbles wins. For precise rules see [1].
Owari is not only a very popular game in certain parts of Africa but has also attracted some interest in the scientiﬁc
community. The strategical aspect was studied by Bal and Romein [2], who have determined by use of computer that the
game is always a draw if both players play optimally. Erickson [7] considered similar games from a more combinatorial
perspective. In contrast, Eglash [5,6] and Bouchet [3,4] focussed on the sowing operation and, in particular, on recurrent
constellations of pebbles under repeated sowing. This will be our main interest too.
Let us make this more precise. A state of the owari is described by the number h of the pit from which we will sow
next, this is called the active pit, and a tuple with the numbers of pebbles in the respective pits. In the sowing move,
we sow from the active pit, which means that we remove all pebbles from pit h and distribute them one by one in
subsequent pits. Then we advance to pit h+ 1, i.e. pit h+ 1 becomes active (here, we take the index h+ 1 modulo the
number of pits). In the next sowing move, we sow from pit h+ 1 and so on. We call a state periodical if after p sowing
moves the number of pebbles in pit i + p equals the number of pebbles in pit i of the starting state, for all i.
 This work was supported by the Algorithmic Discrete Optimization Network (ADONET).
E-mail address: hbruhn@gmx.net.
0012-365X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2007.07.014
H. Bruhn /Discrete Mathematics 308 (2008) 3694–3698 3695
Fig. 1. A period of length 2; the active pit is drawn in bold.
......
Fig. 2. A marching group in an open owari.
......
......
......
Fig. 3. Marching group of order 3, augmented by two pebbles.
A periodical state of period length 2 is illustrated in Fig. 1. We start with the pit to the bottom of the left owari, sow
from it and advance one pit in counterclockwise direction. After two steps we have regained our starting state (modulo
a rotation of the owari).
In Fig. 1 we can witness another phenomenon. Since the ﬁrst pit contains at least as many pebbles as there are pits,
we make a complete tour when sowing from it and thus put a pebble in it.1 Whenever this happens we say that the
sowing overlaps.
In order to avoid overlapping Bouchet considered open owaris. While a closed owari has ﬁnitely many pits arranged
in a cycle as described in the previous paragraphs, an open owari comprises inﬁnitely many pits, laid out so that each
pit has a successor. An open owari may be seen as the limit of closed owaris.
Sowing is done as in closed owaris. An example of a periodical state in an open owari is given in Fig. 2. In contrast,
adding two more pebbles to the ﬁrst (non-empty) pit would yield a non-periodical state. The constellation as in the
ﬁgure is of special interest as it is invariant to sowing; such a state is called a marching group (of an open owari). It is
easy to see that marching groups always have the following form: the ﬁrst non-empty pit is active and contains r ∈ N
pebbles, the pit following it contains r − 1 pebbles, and so on, to the last non-empty pit containing exactly one pebble.
The integer r is called the order of the marching group.
We deﬁne an augmented marching group as follows: starting from a marching group we can decide for every non-
empty pit independently whether to add a pebble or not. In addition, we are allowed (but do not need to do so) to put
one pebble in the ﬁrst empty pit following the marching group. As an example, consider Fig. 3, where the marching
1 We should point out that the standard rules are slightly different. When sowing from pit h it is not allowed to put a pebble in pit h. Rather, if
necessary, pit h is skipped and the pebble put in the next pit. Our rule bending, however, is not substantial. It is not hard to see that there is a natural
bijection between periodical states under standard rules and periodical states under our simpliﬁed rules.
3696 H. Bruhn /Discrete Mathematics 308 (2008) 3694–3698
group of order 3 is augmented by putting one extra pebble in the ﬁrst non-empty pit, and one pebble in the empty pit
succeeding the marching group.
Bouchet [3] characterises the periodical states in an open owari:
Theorem 1 (Bouchet [3]). A state in an open owari is periodical if and only if it is an augmented marching group.
Bouchet also determined the period lengths in an open owari:
Theorem 2 (Bouchet [3]). In an open owari, an integer p is the period of a marching group of order r augmented by
mr pebbles if and only if p = (r + 1)/d where d is divisor of r + 1 and of m.
We prove in the next section an extension of Theorem 1 to closed owaris. Our result will imply Theorem 1. We
remark that our proof is quite different from the one given by Bouchet.
2. Closed owaris
We will deal with closed owaris, which consist of a ﬁnite number, n say, of cyclically ordered pits. We consider the
number n of pits to be ﬁxed. A state of the owari will be represented by a tuple O := [w0, . . . , wn−1] of non-negative
integers, the ﬁrst of which corresponds to the pebbles in the active pit, i.e. the pit from which we will sow next. Let
w0 = qn + r with 0rn − 1. The process of sowing can then be modelled by the sowing operator S, which we
deﬁne as follows: SO := [w′0, . . . , w′n−1] where w′i = wi+1 + q + 1 for i = 0, . . . , r − 1 and w′j = wj+1 + q for
j = r, . . . , n − 1 (setting wn = 0).
Since we are working in a closed owari rather than an open one, we need to amend the deﬁnition of a marching
group. For non-negative integers q, r with rn − 1 we call the state [m0, . . . , mn−1] a marching group (in a closed
owari) and denote it by Mq,r if mi = (n − i)q + r − i for i = 0, . . . , r − 1 and mj = (n − j)q for j = r, . . . , n − 1.
It is easy to see, but will formally be proved in the next lemma nevertheless, that, again, a marching group is invariant
under sowing. Indeed, Mq,r can be seen as the sum of the two marching groups Mq,0 and M0,r . In Mq,0, sowing simply
means putting q pebbles in each pit. On the other hand, as rn − 1, no overlap occurs when sowing in M0,r and so,
M0,r behaves as a marching group in an open owari. See Fig. 4 for an example of a marching group in a closed owari.
Lemma 3. A state in a closed owari is invariant underS if and only if it is a marching group.
Proof. Let W = [w0, . . . , wn−1] be a state in a closed owari, and putSW = [w′0, . . . , w′n−1].
First, letW =Mq,r be amarching group. Thenw′i =wi+1+q+1=(n−(i+1))q+r−(i+1)+q+1=(n−i)q+r−i
for i =0, . . . , r −1 (note that also wi = (n− i)q + r − i for r = i) and w′j =wj+1 +q = (n− (j +1))q +q = (n− j)q
for j = r, . . . , n − 1.
Second, assumeSW=W , and letw0=qn+r where 0rn−1. Thus,wi=w′i=wi+1+q+1 andwj=w′j=wj+1+q
for 0 ir −1 and rjn−1. Starting with wn−1 =q we can easily solve this system of equations, whose solution
is the marching group Mq,r . 
We call a state [m0 + a0, . . . , mr + ar ,mr+1, . . . , mn−1] an augmented marching group (in a closed owari) if
[m0, . . . , mn−1] = Mq,r for some integer q0 and if ai ∈ {0, 1} for all i. As an example, we remark that Fig. 1 shows
Fig. 4. Marching group M2,1 in an owari with 3 pits.
H. Bruhn /Discrete Mathematics 308 (2008) 3694–3698 3697
the marching group M1,1 augmented by one pebble in pit 1. Note that the augmentation of the marching group M0,r
can be viewed as an augmented marching group in an open owari.
A state W of a closed owari is periodical if there is an integer p1 so thatSpW = W . The smallest such p is the
period of W . Marching groups are precisely the states that have period 1.
Let us now ﬁnally state and prove our main result:
Theorem 4. A state in a closed owari is periodical if and only if it is an augmented marching group.
Proof. To check sufﬁciency, let W = [m0 + a0, . . . , mr + ar ,mr+1, . . . , mn−1] be an augmented marching group, i.e.
let Mq,r = [m0, . . . , mn−1] for some q, and let a0, . . . , ar ∈ {0, 1}. Then
SW = [m0 + a1, . . . , mr−1 + ar ,mr + a0,mr+1, . . . , mn−1],
and clearly,SrW = W .
So, let us prove that every periodical state, W = [w0, . . . , wn−1] say, is an augmented marching group. For j0,
put [wj0 , . . . , wjn−1] := SjW . Choose m0 maximal such that there is an integer j0 and a marching group Mq,r =
[m0, . . . , mn−1] with miwji for all i. Put di = wi − mi for all i. Since W is periodical, and since wji − mi0 for
some j, it follows that di0 as well for all i.
We will reduce the sowing operation on W to an operation on the tuple D = [d0, . . . , dn−1]. More generally, deﬁne
an operationSr on a state [e0, . . . , en−1] as follows:
(I) [0, e1, . . . , en−1] → E, e0 → pebbles and r → position
(II) While pebbles > 0 do increase E at position by 1
decrease pebbles by 1
increase position by 1 (modulo n)
(III) if E =[e′0, . . . , e′n−1] then output [e′1, . . . , e′n−1, e′0]
With this operation we get thatSW = M +SrD, where addition is taken componentwise. Since W is periodical
underS so is, therefore, D underSr .
Let dji be such thatS
j
rD = [dj0 , . . . , djn−1]. We will prove the theorem in four steps. First, we claim that
for all integers j0 there is an index i ∈ {0, . . . , r} such that dji = 0. (1)
Suppose that dji 1 for i=0, . . . , r for some j. Putm′0=m0+1, and assume ﬁrst that r <n−1. Then,m′0=qn+(r+1),
andwe putm′i := (n−i)q+(r+1)−i=mi+1 for i=0, . . . , r−1,m′r := (n−r)q+1=mr +1 andm′i := (n−i)q=mi
for i = r + 1, . . . , n − 1. Thus, the m′i form the marching group Mq,r+1. Then, m′i = mi + 1mi + dji = wji for
i = 0, . . . , r and m′i = miwji for ir + 1, contradicting our maximal choice of m0. If r = n − 1 we see in a similar
way that m′0 and the marching group Mq+1,0 would have been a better choice. This proves (1).
Secondly, we show that
there is a constant cr such that for all integers j0 the number cj
of non-zeros among dj0 , . . . , djr is exactly c. (2)
Observe that cj cj+1 for all j. Indeed, sowing or, more precisely, application ofSr can only introduce a new zero at
position r. Since dj+1r = djr+1 + e where e1 if dj0 1, this can only happen if dj0 = 0. But then the gain of the zero
at position r is balanced by the loss of a zero at the ﬁrst position. Thus, cj cj+1. Since D is periodical underSr we
deduce c0 = cj= : c for all j. Claim (1) implies cr .
Thirdly, we claim that
for all integers j0 it holds that dji 1 for i = 0, . . . , r . (3)
Suppose not. Let s be the smallest positive integer for which there exist an integer j and k ∈ {0, . . . , r} such that djk 2
and djk+s = 0 where the index is taken mod (r + 1) (such an s exists, by (1)). By applyingSr k more times, we may
3698 H. Bruhn /Discrete Mathematics 308 (2008) 3694–3698
assume that k = 0. Thus,SjrD starts with an element 2, which is followed by s entries of 1, which, in turn, precede
a 0. Assume ﬁrst that r <n − 1. Then, further applications ofSr yield the following:
[2,
s times
︷ ︸︸ ︷
1, . . . , 1, 0, ∗, . . . , ∗] → [
s times
︷ ︸︸ ︷
1, . . . , 1, 0, ∗, . . . , ∗,
pos r
1 , 1, ∗, . . . , ∗]
→ [
s−1 times
︷ ︸︸ ︷
1, . . . , 1, 0, ∗, . . . , ∗, 1,
pos r
2 , ∗, . . . , ∗]
→ s−1 applications of Sr. . . →
→ [0, ∗, . . . , ∗, 1, 2,
s−2 times
︷ ︸︸ ︷
1, . . . , 1,
pos r
1 , ∗, . . . , ∗]
→ [∗, . . . , ∗, 1, 2,
s−1 times
︷ ︸︸ ︷
1, . . . , 1,
pos r
0 , ∗, . . . , ∗].
The entry of 0 at position r in the last state cannot be larger than zero, since in that case cj would increase, which
is impossible, by (2). Then, however, there at most s − 1 ones between an entry of 2 and the zero at position r,
contradicting our choice of s. For r = n − 1 the argumentation is similar. This proves (3).
In order to ﬁnish the proof of the theorem it sufﬁces, in view of (3), to establish the following claim:
dr+1 = · · · = dn−1 = 0. (4)
Suppose there is an ir + 1 for which di1. Then dj+1r di + e for j = i − r and e= 1 if dj0 1 and e= 0 otherwise.
From (3) it follows that e = 0 as otherwise dj+1r 2. However, e = 0 and thus dj0 = 0 entails the introduction of a new
non-zero among dj0 , . . . , d
j
r in step j + 1, i.e. cj < cj+1, which contradicts (2). 
We note that Theorem 1 follows easily from our main result. Indeed, let Q be a ﬁnite tuple representing consecutive
pits of a periodical state in an open owari such that every pit outside this range is empty, and let n − 1 be the total
number of pebbles the state contains. By appending 0-entries we can turn Q into a state W of a closed owari with n pits.
Because of the size of the owari an overlap will never occur, and W will be periodical too. Therefore, W is a marching
group Mq,r augmented by m pebbles. As W contains only n pebbles, it follows that q = 0. Since M0,r can be seen as a
marching group in an open owari it follows that Q represents an augmented marching group in an open owari.
With the same reduction we can use Theorem 1 to characterise the periods of closed owaris too.
Acknowledgement
I would like to thank Attila Bernáth for helpful discussions.
References
[1] 〈http://www.oware.org〉.
[2] H.E. Bal, J.W. Romein, Solving the game of Awari with parallel retrograde analysis, IEEE Comput. 36 (2003) 26–33.
[3] A. Bouchet, Owari I. Marching groups and periodical queues, 2005, Preprint. 〈http://www.rpi.edu/eglash/isgem.dir/texts.dir/OwariI.pdf〉.
[4] A. Bouchet, Owari II. Marching groups and Bulgarian Solitaire, 2005, Preprint 〈http://www.rpi.edu/eglash/isgem.dir/texts.dir/OwariII.pdf〉.
[5] R. Eglash, African Fractals: Modern Computing and Indigenous Design, New Brunswick, NJ: Rutgers University Press, 1999.
[6] R. Eglash, L’algorithme ethnique, Dossier Pour la Science 47 (2005) 102–104 (in French).
[7] J. Erickson, Sowing games, in: R.J. Nowakowski (Ed.), Games of No Chance, New Brunswick, NJ: MSRI Publications, 1997, pp. 287–297.


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Periodical states and marching groups in a closed owari

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