A028932 Triangular array of winning positions in Tchoukaillon (or Mancala) solitaire.

0, 1, 2, 0, 2, 1, 3, 1, 0, 3, 1, 1, 4, 2, 0, 0, 4, 2, 0, 1, 4, 2, 2, 0, 4, 2, 2, 1, 5, 3, 1, 1, 0, 5, 3, 1, 1, 1, 6, 4, 2, 0, 0, 0, 6, 4, 2, 0, 0, 1, 6, 4, 2, 0, 2, 0, 6, 4, 2, 0, 2, 1, 6, 4, 2, 3, 1, 0, 6, 4, 2, 3, 1, 1, 7, 5, 3, 1, 2, 0, 0, 7, 5, 3, 1, 2, 0, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• easy
• nice

Comments

The 300 names of Mancala: Abalala'e, Abanga, Abangah, Abouga, Achara, Adi, Adita ta, Adito, Adji, Adjiboto, Adjika, Adji pre, Adjito, Aghi, Agi, Aji, Ajwa, Ale, Andot, Annana, Anywoli, Awale, Awale, Aware, Awari, Awele, Awele Ayo, Ayo Ayo, Azigo,

Ba-Awa, Banga, Bao, Bao kiswahili, Bao solo, Bare, Baruma, Bau, Bawo, Bechi, Boke, Bosh, Bouberoukou, Bouri, Chanka, Chisolo, Chongkak, Choro, Chouba, Chuba, Chunca, Cisolo, Congkak, Coo, Coro, Coro bawo,

Dabuda, dakon, Dakoun, Dara, Darra, Deka, Djonghok, Djonglak, Dwong, erherhe, Endodoi, Enkeshui, Eson xorgol, Esson, eu leu, Fangaya, Fuva, Gabata, Gabatta, Galatjang, Gamacha, Gbegele, Gebta, Gelo, Gepeta, Gesuwa, Gilberta, Giuthi,

Halusa, Hus, Igisoro, Ikiokoto, Imbelece, Imbwe, Impere, Isafu, Ise onzin egbe, Isofu, Isolo, J'erin, jodu, J'odu, Jukuru, Kachig, Ka ia, Kalah, Kalaha, Kalak, Kale, Kalimanta, Kasonko, Katra, Kboo, Kenji guki, Kiarabu, Kisolo, Kiswahilibao, Kiuthi, Kpo, !Krour, Kubuguza,

La'b hakim, La'b madjunni, La'b roseya, Lahemay walida, Lami, Lamlameta, Lamosh, Lam waladach, Langa holo, Layo, leka, Lela, Leyla gobale, Lien, Lizolo, L'ob akila, Lonbeu a cha, Lontu Holo, Luzolo,

Mancala, Mandiare, Manga, Mangala, Mangola, Mankala, Manqala, Manquala, Marabout, Marany, Maruba, Mazageb, Mbangi, Mbau, Mbelete, Mbere, Mbo, Mbothe, Mefuhva, Meusueb, Mewelad, Mofuba, Moro gbegele, Motiq, Msuwa, Mulabalaba, Mungala, Mutiteba, Mwambalula, Mweiso, Mweso,

Nakabile, Nambayi, Naranj, Ncholokoto, Nchomvwa, Nchuba, Nchuwa, Ndoto, Ngar, Njombwa, Nocholokoto, Nsolo, Nsumbi, Ntchuwa, Nummun,

Oko, Olinda, Okwe, Omweeso, Omweso, Otep, Otjitoko, Ot jun, Otra, Ot tjin, Otu, Oure, Ouri, Ourin, Ourre, Ourri, Oware, Owela, Palankuli, Pallamkurie, Pallam kuzhi, Pallanguli, Pallankuli, Pandi, Papadakon, Papandata, Pensur, Pereauni, Peresouni, Poo, Qaluta, Qasuta, Qelat, Ryakati,

Saddeka, Sadeka, Sadiqa, Schach, Serata, Sig, Solo, Sombi, Songo, Soro, Spreta, Sulus ni!shtaw, Sunca, Sungka, Tagega, Tamtam apachi, Tap, Tapata, Tchanka, Tchokajon, Tchonkkak, Tchoukaitlon, Tchukaruma, Tegre, Tjonglak, Toguz xorgol, Toi, Tonka, Topuz xorgol, Tchuba, Tchela, Tshuba, Tshi solo, Tsoro,

Ubao, Ugwasi, Um el bagara, Um el banat, Um el tuweisat, Urdy, Ure, Vai lung thlan, Wale, Walle, Walu, Walya, Ware, Wari, Warri, Wawee, Wawi, Weg, Woaley, Wori, Woribo, Woro, Wouri, Wuli, Wuri, Xorgol, Yada, Yit nuri, Yovodji.

Examples

			Triangle begins:
0,
1,
2, 0,
2, 1,
3, 1, 0,
3, 1, 1,
4, 2, 0, 0,
4, 2, 0, 1,
4, 2, 2, 0,
4, 2, 2, 1,
5, 3, 1, 1, 0,
5, 3, 1, 1, 1,
6, 4, 2, 0, 0, 0,
6, 4, 2, 0, 0, 1,
6, 4, 2, 0, 2, 0,
6, 4, 2, 0, 2, 1,
6, 4, 2, 3, 1, 0,
6, 4, 2, 3, 1, 1,
7, 5, 3, 1, 2, 0, 0,
7, 5, 3, 1, 2, 0, 1
...
		

References

• Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11 (1957), 26-31.
• C. Zaslavsky, Africa Counts: Number and Pattern in Traditional African Culture, Boston: Prindle, Weber and Schmidt, 1973.

Links

• D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
• E. Brisse, African Games Information and Links
• D. M. Broline and _Daniel E. Loeb_, The combinatorics of Mancala-Type games: Ayo, Tchoukaillon and 1/Pi, J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.
• D. Eppstein, Combinatorial Game Theory
• P. Erdős and E. Jabotinsky, On a sequence of integers generated by a sieving process (Part I), Indagationes Math., 20, 115-123, 1958.
• P. Erdős and E. Jabotinsky, On a sequence of integers generated by a sieving process (Part II), Indagationes Math., 20, 124-128, 1958.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• R. W. Wilder, Review of "Africa Counts: Number and Pattern in Traditional African Culture", Historia Mathematica, Vol. 2 (1975), pp. 207-210.
• C. Zaslavsky, African and Multicultural Mathematics
• Index entries for sequences generated by sieves

Crossrefs

Cf. A002491, A007952, A028920, A028931, A028933.

Programs

Mathematica

s[list_] := Module[{x = Append[list, 0], i = 1}, While[x[[i]] =!= 0, x[[i]] = x[[i]] - 1; i = i + 1]; x[[i]] = i;If[Last@x == 0, Most[x], x]]; Flatten[Reverse /@ NestList[s, {}, 20]] (* Birkas Gyorgy, Feb 26 2011 *)

Formula

To get the next row in the triangle, find the rightmost zero entry in the current row (which may be to the left of the existing entries). In this zero is in position k (counting from the right), change it from 0 to k and subtract 1 from all the entries to its right.

Extensions

Additional references from Labos Elemer, Nov 07 2000

Revised by N. J. A. Sloane, Jul 16 2012