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Chihwangu occurs for: 2N holes per row and (2N)-1 pebbles in each hole.
This seems to be just a special case of a more general hypothesis:
Chihwangu occurs for: N holes per row and N-1 pebbles in each hole, for all N > 3.
For the proof of this hypothesis I will use notation as depicted in the figure:
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In this notation, holes are numbered from 1 to N in the playing direction, and restarting for each row. A distinction between the inner row and outer row will be made by preceding the hole number by an ``I" for the inner row, or an ``O" for the outer row. If it needs to be, a distinction between the two players could be made by preceding that key again with an ``N" for the north player or an ``S" for the south player. For the proof of the Chihwangu hypothesis, the north/south indication is omitted since Chihwangu needs only 1 turn, where it does not matter whether North or South makes this move.
The proof:
The proof that Chihwangu occurs for boards with N holes per row and N-1 pebbles per hole (from here on I will call this an ``(N, N-1) board") is based upon my observation that for such boards always the same hole is involved with Chihwangu. For the proof, we will simply count out the distribution of the pebbles over the board. If we can continue this counting until all pebbles are on our side of the board without having set any restriction to N, we have proven that the hypothesis is valid for any (N, N-1) board.
We split the proof in two separate cases: N is even, and N is odd.
Sowing no. 2: The 2N-2 pebbles are distributed over our own holes, N
on
the outer row, continuing with N-2 pebbles on the inner row. Sowing 2
ends therefore in hole I(N-2).
Capture no. 2: The pebbles from the opposing player's I(3) and O(N-2),
giving a total of 2×(N-1) pebbles.
Sowing no. 3: The 2N-2 pebbles are distributed over our own holes, 2 on
the inner row, N on the outer row, leaving N-4 for the inner row.
Sowing 3 ends in hole I(N-4).
Capture no. 3: 2×(N-1) pebbles from the opposing player's I(5) and
O(N-4).
It can easily be seen that every next capture occurs two holes to the right, until finally:
Capture no. N/2: 2N-2 pebbles from the opposing player's I(N-1) and O(2). These 2N-2 pebbles are NOT enough to walk around to our own inner row again! The sowing ends in hole O(N). At this point, hole O(N) contains \dsN-1+(N/2) pebbles (the initial N-1 pebbles plus \dsN/2 pebbles that are sown). So we take those \dsN-1+(N/2) pebbles, and continue sowing:
Sowing no. \ds(N/2)+1: N pebbles on the inner row, leaving \ds(N/2)-1 pebbles for the outer row. This sowing ends in hole O(\ds(N/2)-1), which now contains the initial N-1 pebbles plus \ds(N/2)+1 pebbles from sowing. We take those \dsN-1+(N/2)+1=N+(N/2) pebbles and continue sowing.
Sowing no. \ds(N/2)+2: Since the previous sowing ended in hole O(\ds(N/2)-1), we still have N-(\ds(N/2)-1) holes on the outer row to sow, which leaves for the inner row: \dsN+(N/2)-(N-((N/2)-1)) = N-1 pebbles. So this sowing ends in hole I(N-1) after which we can again capture 2×(N-1) pebbles from the opponent's holes I(2) and O(N-1)!
From this point on every next capture occurs two holes to the right again, until finally all the opponent's pebbles are captured, hence: Chihwangu!
Sowing no. \ds((N+1)/2)+1: N-1 pebbles on the inner row, and N-1 pebbles on the outer row. The sowing ends in hole O(N-1). So far we have made \ds((N+1)/2) captures and distributions, therefore this hole now contains N-1 + \ds((N+1)/2) pebbles. We take those N-1 + \ds((N+1)/2) pebbles and continue:
Sowing no. \ds((N+1)/2)+2 : 1 pebble on the outer row, N pebbles on the inner row, leaving N-1+\ds((N+1)/2)+2 - 1 - N = \ds((N+1)/2)-2 pebbles for the outer row. The sowing ends in hole O(\ds((N+1)/2)-2), which contains one more pebble than at the end of the previous sowing, therefore N-1 + \ds((N+1)/2)+1 = N+((N+1)/2) pebbles).
Sowing no. \ds((N+1)/2)+3: There are N-(\ds((N+1)/2)-2) holes left on the outer row. We start this sowing with N+\ds((N+1)/2) pebbles in the hand, so we have for the inner row: N+\ds((N+1)/2) - (N-(((N+1)/2)-2)) = (N+1)-2 = N-1. Therefore this sowing ends in hole I(N-1), and we capture 2×(N-1) stones again!
From here on every next capture is made two holes to the right again, until the last hole, and: Chihwangu!
In the above proof, no limitation is set to the number N, therefore the hypothesis should be valid for any N. Yet, in practice the hypothesis seems not to be valid for N < 4. Can this be argued somehow? Let's see: of course N=0 (no holes at all) and N=1 (one hole per row, but no pebbles) are meaningless, but what about N=2 and N=3?
For N=2 we have to look at the proof for even numbers: at the start of sowing \ds(N/2)+1 we have \dsN-1+(N/2) pebbles in the hand, which would be 2 pebbles for N=2. We sow them out, and end the sowing (according to the proof) at hole O(\ds(N/2)-1), but for N=2 the sowing ends on the inner row, not the outer row! In order to end sowing no. \ds(N/2)+1 on the outer row, the proof should have the assumption (or limitation) N-1+\ds(N/2) > N (number of pebbles in hand > number of holes per row), which reduces to N > 2, so indeed: the proof is invalid for N=2.
For N=3 we look at the proof for odd numbers: sowing no. \ds((N+1)/2)+2 is assumed to start and end on the outer row, but in order to achieve that, the pebbles in the hand must be more than the holes left on the outer row plus the whole inner row, so: N-1+\ds((N+1)/2) > 1+N, which reduces to N > 3, so indeed the proof is invalid for N=3. Note that for N=3 Chihwangu occurs if we begin with hole I(3) instead of I(1).
The final conclusion: Chihwangu occurs for (N, N-1) boards, for any N > 3. The hole to be played to achieve Chihwangu is I(1).
However, I have observed that for many boards Chihwangu can be achieved by playing other holes too. In fact, (N, N-1) boards (with N > 3), always have at least one other hole which leads to Chihwangu. This other hole is not fixed however, but varies with the board. Unlike my earlier hypothesis, I cannot prove this one (it would be interesting if one of the Z Zimaths readers could come up with a proof).
I have some more observations about this intriguing game, which I will list below. Please note that they are all just observations, without any proof. In fact, these observations might even be proven wrong! I challenge all Zimaths readers to come up with a proof either confirming or denying my observations.
Observations:
1. For Tsoro boards of the form (N, N-1) with N > 3, the first hole of
the inner row always leads to Chihwangu (this is the observation which I
used in my earlier article to proof that such boards always lead to
Chihwangu).
2. For Tsoro boards of the form (N, N-1) with N > 3, there are always at
least two holes leading to Chihwangu.
3. For Tsoro boards of the form (N, N-1) with N > 3, always two of the
holes leading to Chihwangu are corner holes (with this I mean that these
holes are located on the edges of the current players side of the board).
4. There is no single Tsoro board of the form (N, M) for any N and any
M,
for which there are more than 4 holes leading to Chihwangu.
The observations above are based on a lot of experiments with various board sizes and amounts of pebbles. The results are listed in the matrix below. In any cell of the matrix, holes which lead to Chihwangu are listed .
To keep the notation of these holes as short as possible, I use just one letter to indicate a hole. This in contrary to the notation used in the earlier mentioned article, where I numbered the holes. The lettering is as follows: the first hole in the inner row gets the letter a, the second the letter b, etc. The holes on the outer row receive the same letter as their opposing inner hole, but capitalized. Therefore the last hole on the outer row gets the letter A, the last-but-one gets the letter B, etc. For practical reasons, I limited my research to boards with max. 20 holes per row, and max. 20 pebbles per hole. I then come to the following Chihwangu matrix, with horizontally the number of holes per row and vertically the number of pebbles per hole:
In case readers want to experiment with Tsoro themselves, I would like to
point to my website where I have placed a JavaScript version of the Tsoro
game. There is a game page where visitors can play against the computer
(with the possibility to be added to a high score list if one manages to
win a game), and there is an experimental page where various parameters
of the game like board size, pebbles per hole, and some of the rules can
be changed, to investigate what impact this has on the game itself.
The URL of the Tsoro website is given at the end of the next article.
I live in Apeldoorn, the Netherlands. Although I am a graduate
electronics engineer, I am currently working as software engineer for a
company that develops and installs radar equipment and software for
airports and harbors.
I first learned about the game of Tsoro during a holiday visit to
Zimbabwe in 1996. In a shop near Victoria Falls I saw a cut wooden board
with dried beans on it. I asked the shopkeeper what it was, and he told
me that it was a game (which I already guessed). When I asked him if he
could explain the rules to me, he did so and also gave me a short demo
game. I bought the board.
Back home I played the game several times against my brother, but he
finally lost interest while I became addicted. I started a search for
information about Tsoro on the Internet. I expected to find some
references to the game and possibly a few playable programs too. To my
surprise, I was not able to find anything about Tsoro on the entire World
Wide Web. So I decided to write myself a Tsoro-playing computer program,
in order to be able to play Tsoro whenever I wanted. That program worked
fine, but was graphically very simple. It used plain numbers instead of
graphics to indicate the amount of stones on a field.
Some years later (I taught myself JavaScript in the mean time) I decided
to develop a JavaScript version of Tsoro, to share the game with the
world. By the time that I put that game on the web, I retried my search
for information about Tsoro. This time I found exactly one reference,
being Issue 3.2 of Z Zimaths, where Mr. Thomas Masiwa explained a version
of Tsoro which was a bit different than mine. I asked the editors of
Z Zimaths if they could bring me into contact with Mr. Masiwa, but heard
nothing in response. Almost a year later, to my surprise I received an
email from Mr. Masiwa! We had some email discussions about Tsoro
(especially about his investigation of Chihwangu), and he invited - or
maybe challenged - me to write an article for Z Zimaths about Tsoro. I
accepted the invitation, and the result is what you are currently looking
at.
My Tsoro website has gained some popularity lately. It has been game of
the month on the website of a well known American pizza restaurant, and
it is being linked by an American hotel chain as part of their internet
entertainment program. There are enthusiastic players from all over the
world that return to my page on a regular basis (possibly to see if their
entry in the Hall Of Fame has been beaten), and lately the Tsoro page has
been visited a lot by a group of Zimbabwean students in the U.S.A. .
Inspired by my website, one of these students has decided to write a
Tsoro-playing program as a final project of his Computer Science study.
The URL of the website is: http://home.hetnet.nl/ wasse257/tsoro/
The accompanying email address is: tsoro_thegame@hotmail.com
pebs. H o l e s p e r r o w per hole 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 c b 3 aA 4 C ae c 5 aBA D aA 6 agGA 7 aB cA aA 8 a ai a 9 b bC aIA 10 abcC akA L 11 abBA C aA 12 am 13 aA B abMA 14 c aoOA 15 abB dD aA 16 b A aqQ 17 a dD f A aA 18 asA 19 A B D beE f a aA 20 c gG
3 My love affair with Tsoro: a personal note
by Lex Wassenberg
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