Pyramids (九塔棋)

Man fears time, but time fears the Pyramids.
Arab proverb

PYRAMIDS (九塔棋)
3rd edition
revised and improved

Kosintsev I.G.
kosintsev.i.g@gmail.com
(10.09.2012)

When you are travelling near Egypt's, China's or Mexico’s pyramids you may ask yourself: what game are Gods playing?
Albert Einstein once said: God does not play dice with the universe. Yes! Of course! Because Gods play pyramids!
You can also try it by the new similar to chess game entitled “Pyramids”.

The initial position of the game Pyramids for advanced players (the standard 9x9 board)


Definition of a pyramid.
Vertically adjacent checkers (particles) of the one color with different sizes (in oure case of small C1 to large C4) and with the correspondent numbers (labels) form a pyramidal labeled polyomino or partially ordered multiset (simply called "pyramid"). It is clear that pyramid has at least one checker.

The number of all possible pyramids((pieces) is restricted to 49 by the next rule: any pyramid (P) can consist of no more than four checkers and can have no more than two (double) checkers of one type (see Appendix).

You can see all possible pieces (piramids, slave) of the game Pyramids on the next diagramm.



It is possible to reprisent any position by the digital consequences instead of pieces and by Algebraic chess notation.

The list of all pieces and the conventional designations (notation of all 49 Pyramids in nonary [5]:

(without number) S slave

1. P(1) turtle (king of chess [6])
2. P(11) lizard
3. P(2) viper (knight of chess [6])
4. P(21) cobra
5. P(211) python (squirrel of chess [6])
6. P(22) boa
7. P(221) anaconda
8. P(2211) alligator

10. P(3) meduse
11. P(31) octopus
12. P(311) eel
13. P(32) skate
14. P(321) pike
15. P(3211) shark
16. P(322) dolphine
17. P(3221) whale

20. P(33) cock (bishop of chess [6])
21. P(331) gull
22. P(3311) crane
23. P(332) parrot (cardinal of Capablanca chess [6])
24. P(3321) owl
26. P(3322) raven

30. P(4) beaver
31. P(41) sable
32. P(411) squirrel
33. P(42) hare
34. P(421) fox
35. P(4211) wolf
36. P(422) panda
37. P(4221) bear

40. P(43) macaque
41. P(431) chimpanze
42. P(4311) orangutan
43. P(432) baboon
44. P(4321) pharaon
46. P(4322) gorilla

50. P(433) falcon
51. P(4331) condor
53. P(4332) eagle

60. P(44) goat (rook of chess [6])
61. P(441) horse
62. P(4411) giraffe
63. P(442) bull (marshal of Capablanca chess [6])
64. P(4421) rinoceros
66. P(4422) elephant

70. P(443) panther
71. P(4431) tiger
73. P(4432) leon

80. P(4433) dragon (queen of chess [6])

The aim of the game is to remove all opponent's pharaons P(4321) off the board by a consequence of alternate (white and black) moves according to the rules.

Types of movements of pieces.

Type M. “March” – the simple move without capturing.
In that case one whole pyramid moves from one square to another unoccupied (free) square of own destination.

Type C. “Capture” – the move with capturing.
In that case one whole pyramid moves from one square to another, occupied by the opponent’s pyramid which is removed off the board.

Type P. “Partition” – the partition of one hole pyramid.
Any pyramid may be subdivided, and in effect becomes two different (by disposition on board) pyramids.
In that case only part of pyramid (subset of checkers or not all checkers) moves from one square to another unoccupied (free) square, another part of this pyramid remain unmoved.

Type A. “Amalgamation” (Junction) – the junction of two different (by disposition on board) pyramids in one whole.
A pyramid may be expanded by moving additional checkers on free vacancies, if the produced pyramid is legal.
In that case one whole pyramid moves to square of destination occupied by another own (the same color) pyramid and all checkers of two pyramids compound the new pyramid.

Attention! Any movement of a mixed (hybrid) type is forbidden.

Modes of movements of pieces.

1. The orthogonal mode of the movement O or O(n), where n is the number of the path cells.

2. The diagonal mode of the movement D or D(n), where n is the number of the path cells.

3. The first, second ant third neighbor mode of the movement N(n), where n=1,2,3 over the neighbor occupied or unoccupied squares N(x), where x less n (Jump).

The classical variant of Pyramids has such rules for movement of pieces.
1. A slave (S) moves only ahead by N(1) (i.e. O(1) or D(1))mode with or without capturing.
It also promotes for any another (at the player's choice) pyramid when it reaches the last rank.
2. Any pyramid (P) with single checker C1 moves by N(1) mode.
3. Any pyramid (P) with double checker C1 and without double checker C2 jumps by O(2) or D(2) mode.
4. Any pyramid (P) with single checker C2 moves by N(2) mode without jumps by O(2) or D(2) mode.
5. Any pyramid (P) with double checker C2 and without double checker C1 moves by N(3) mode without jumps by O(3) or D(3) mode.
6. Any pyramid (P) with double checker C1 and with double checker C2 moves by N(3) mode.
7. Any pyramid (P) with checker C3 moves by D(1) or D(2) or D(3) mode, with double checkers C3 - by D mode. If also with single (double) checker C1 then can jump over D(1) (D(1) and D(2)) square(s).
8. Any pyramid (P) with checker C4 moves by O(1) or O(2) or O(3) mode, with double checkers C4 - by O mode. If also with single (double) checker C1 then can jump over O(1) (O(1) and O(2)) square(s).
9. The pharaon P(4321) moves by N(1) mode.

The cheskers (chess & checkers) variant of Pyramids has own rules for movement of pieces and own particular rules of a capture.
If one player's piece, only one other player's piece, and an empty square are lined up, then the first player may "jump" the other player's piece. In this case, the first player jumps over the other player's piece onto the empty square and takes the other player's piece off the board.
If the piece can jump with the move, it must jump. Sometimes a player may have the option or a choice of which opponent piece he must jump. In such cases, he must then choose which to jump.
1. A slave (S) (moves only ahead) marches without capturing by OD(1) mode and jumps by OD(2) mode.
It also promotes for any another (at the player's choice) pyramid when it reaches the last rank.
2. Any pyramid (P) with checker C1 marches by D(1) mode and jumps by D(2) mode.
3. Any pyramid (P) with checker C2 marches by O(1) mode and jumps by O(2) mode.
4. Any pyramid (P) with checker C3 marches and jumps by D mode.
5. Any pyramid (P) with checker C4 marches and jumps by O mode.
6. Any pyramid (P) with checker C3 or C4 jumps over and hence capture an opponent piece some distance away and choose where to stop afterwards.
7. A player can use one pyramid (P) with double checkers (C1 or C2 or C3 or C4) to make multiple jumps in any one single move, provided each jump continues to lead immediately into the next jump.
8. A player can use one pyramid (P) with checkers (C1 and C3) or (C2 and C4) to make double jumps by each one checker in any one single move, provided first jump continues to lead immediately into second jump.
9. The pharaon P(4321) marches by N(1) mode and jumps by OD(2) mode.
.
Other common rules of these games are identical with rules of all board games (chess, draughts et al.).

Internet resources:
1. Chess.
http://en.wikipedia.org/wiki/Chess
2. Chaturanga.
http://www.chessvariants.org/historic.dir/chaturanga.html
3. В. Ивановский, О. Свирин
Русские шахматы: Таврели.
Москва, "Русский путь", 2002 (104 с.)
http://tavreli.narod.ru/books.html
4. Shogi.
http://en.wikipedia.org/wiki/Shogi
5. Nonary.
http://en.wikipedia.org/wiki/Nonary
6. Fairy chess piece.
http://en.wikipedia.org/wiki/Fairy_chess_piece
7. Cheskers.
http://en.wikipedia.org/wiki/Cheskers
8. Draughts.
http://en.wikipedia.org/wiki/Draughts
9/ Richard P. Stanley Enumerative Combinatorics.

Appendix. The number of all possible pyramids.

Let's count the number of all possible pyramids, creatures or simply sets if you are correspondingly constructor, biologist or mathematician).
Let n is the number of types of checkers (in our case n=4) and m is the number of checkers of one pyramid.
In that case we have:
L(m,n,)=C(n-1,n-1+m),
where L is combination with repetition and C is combination.

1) for one checker (m=1)
L(1,4)=C(3,4)=4

P(1)
P(2)
P(3)
P(4)

2) for two checker (m=2)
L(2,4)=C(3,5)=C(2,4)+C(1,4)=10

P(21)
P(31)
P(32)
P(41)
P(42)
P(43)

with doublet
P(11)
P(22)
P(33)
P(44)

3) for three checkers (m=3)
L(3,4)=C(3,6)=C(3,4)+V(2,4)+C(1,4)=20
where A is variations

P(321)
P(421)
P(431)
P(432)

with doublet
P(211)
P(221)
P(311)
P(322)
P(331)
P(332)
P(411)
P(422)
P(433)
P(441)
P(442)
P(443)

unused with triplet
P(111)
P(222)
P(333)
P(444)

4) for four checkers (m=4)
L(4,4)=C(3,7)=C(4,4)+C(2,4)+2V(2,4)+C(1,4)=35

P(4321)

with doublet
P(3211)
P(3221)
P(3321)
P(4211)
P(4221)
P(4311)
P(4331)
P(4322)
P(4332)
P(4421)
P(4431)
P(4432)

P(2211)
P(3311)
P(3322)
P(4411)
P(4422)
P(4433)

unused with triplet
P(2111)
P(2221)
P(3111)
P(3222)
P(3331)
P(3332)
P(4111)
P(4222)
P(4333)
P(4441)
P(4442)
P(4443)

unused with quartet
P(1111)
P(2222)
P(3333)
P(4444)

Total T=L(1,4)+L(2,4)+L(3,4)+L(4,4)=69
Used subset S=T-2C(1,4)-V(2,4)=49

Let's try to count the number of partitions of every pyramid and to build the latice of the set of all pyramids with their partitions.