Xiang Shi Fu
This page is based on a contribution from Dmytro Polovinkin with help from Min Fanxin (闵凡信).
This is a solitaire Chinese domino game in which the aim is to rearrange the layout of 32 tiles into 10 valid triplets and one valid pair. The name Xiàng Shí Fù means "Watching Ten Sets". The game is also known as Cān Chán (參禪) which means Zen.
The Tiles and Valid Sets
A standard 32-tile Chinese domino set is used. The tiles show all 21 different combinations of the numbers 1 to 6. Eleven of the combinations (sometimes known as civil) appear on two identical tiles while the other ten combinations appear on only one tile (military). The aim is to arrange the 32 tiles into ten valid triplets and one valid pair.
As in many Chinese domino games, there are 16 valid pairs that fall into three types.
- Two identical (civil) tiles. The eleven possibilities are pairs of: [6:6] [1:1] [4:4] [3:1] [5:5] [3:3] [2:2] [6:5] [6:4] [6:1] [5:1].
- Two unique (military) tiles whose pip totals are equal. There are four of these pairs: [6:3]&[5:4] (9), [6:2]&[5:3] (8), [5:2]&[4:3] (7), [4:1]&[3:2] (5).
- The 'supreme pair': [1:2]&[2:4].
There are nine types of valid triplet, based on the combinations of six numbers on the six tile-ends: it does not matter how the numbers are distributed across the three tiles. The first three are triplets that include three equal ends.
- Five points. Exactly three equal ends, and the pips on the other three ends add up to 5. Example [3:3][3:1][2:2]. Note that there cannot be more than three equal ends, so for example [3:3][3:1][3:1] and [1:1][1:1][2:2] are not valid triplets.
- Full fourteen. Exactly three equal ends, and the pips on the other three ends add up to 14 or more. Example [1:6][1:6][1:4]. Note that as with 'five points' there cannot be more than three equal ends, so triplets such as [5:5][6:5][5:4] are not valid.
- Split. Two sets of three equal ends. Example [1:1][1:6][6:6].
The next two require four or five equal ends.
- Coincidence. Four equal ends, and the total of the pips on the other two ends also equals the same number. Example [5:5][5:3][5:2], having four fives coinciding with 3+2 which is also five.
- Five sons. Five equal ends. Example [1:1][1:1][1:3].
Each of the last four requires a spread of single numbers or pairs of numbers.
- All different (or Full dragon). All ends are different - one of each number 123456. Example [1:3][2:6][4:5].
- Small dragon. Two of each of the small numbers 1, 2 and 3. Example [1:3][1:3][2:2].
- Big dragon. Two of each of the large numbers 4, 5 and 6. Example [4:4][6:5][6:5].
- 2-3-Kao. Two 2's, two 3's and two 6's. There are only two ways to make this: [2:2][3:3][6:6] or [2:3][2:6][3:6].
Rules of Play
The tiles are shuffled and arranged at random into ten sets of three and a set of two. From the point of view of the game play it does not matter how they are arranged, but the traditional layout is with the pair in the centre surrounded by the triplets like this.
A move is to exchange one or two tiles between two sets, so that after the exchange both sets are valid. The player makes a series of moves aiming to eliminate all invalid sets.
For example in the illustration above the [1:6] in the top right triplet could be exchanged with the [2:3] lower down the right side to create [1:1][2:3][2:2] (five points) and [1:5][1:6][1:4] (full fourteen).
Note that there is no real point in exchanging two tiles between triplets, since exchanging the single third tiles would have an equivalent effect. However, it can be useful to exchange the whole central pair with two tiles from a triplet that form a valid pair.
For example in the illustration above the central pair could be exchanged with the two [6:5]'s on the right to make the pair [6:5][6:5] and [4:4][1:1][1:6] on the right (full fourteen).
The game is won when the layout contains ten valid triplets and one valid pair.
Game with pre-defined target
This solitaire is not easy to solve from a random position: the requirement that every move must create only valid sets is quite restrictive. Nevertheless, to make it even harder some players like to choose before laying out the tableau a specific central pair that must be created together with 10 valid triplets in order to win.
Min Fanxin proposed two other variants with predefined target combinations
- Five dragons fighting for throne (五龙争位). To win the player needs to end with the pair [2:2][2:2], five 'Full dragons' (123456) and five 'Five sons' (triplets with five equal ends).
- Generals in mutual harmony (将相和). To win the player must end with the Harmony pair [1:3][1:3] and every triplet should not only be valid but should also consist of a pair of identical (civil) tiles plus one unique (military) tile.
Two tiles, the [2:4] and one of the [4:6] tiles, are removed, leaving a set of 30 dominoes, which are randomly arranged in triplets at the start of the game. The moves and the valid combinations are the same as in the 32-tile game described above, but the winning position is strictly defined.
In order to win, the player must end up with 4 Full Dragons (123456), 2 Small Dragons (112233), 2 Big Dragons (445566), 1 Silver Ingot and 1 Gold Ingot. The Silver and Gold Ingots are the special cases of the Five Points triplet: the Gold Ingot consists of two [6:1] tiles plus the [6:3] and the Silver Ingot consists of two [5:1] tiles plus the [5:3].
Eight Triplets Variant
The 32 tiles are randomly arranged into 4 pairs and 8 triplets. The valid pairs, triplets and moves are as usual but in order to win the player must end with either the top four civil pairs [6:6][6:6], [1:1][1:1], [4:4][4:4], [3:1][3:1] or the top four military pairs [6:3][5:4], [6:2][5:3], [5:2][4:3], [4:1][3:2] plus any eight valid triplets.
Dead group problem and solutions
Using the classic rules explained above there are certain triplets that cannot be converted to a valid triplet or pair by any exchange. There are five such triplets: [1:4][2:4][3:3], [1:4][2:4][4:5], [1:5][2:5][3:3], [1:5][2:5][4:4] and [2:3][2:4][3:4]. These are known as dead groups because if the initial layout contains any of these triplets the game is impossible to win.
Min Fanxin has explained several solutions that have been proposed by various people to avoid having to abandon games where these groups appear.
- Yangzhou Yue Zhengrong (扬州岳正荣老师) proposed to replace Full Fourteen by the combination 'Three same and Thirteen points', which is like Full Fourteen except that the threshold for a valid combination is lowered to 13 points. With this change four of the classically dead groups can be converted to valid triplets, but one triplet [1:4][2:4][3:3] remains dead.
- Li Shaoquan (李绍泉老师) proposed to replace Two-Three-Kao (223366) by a new combination 'Mixed Dragon' which has one 2, two 3's, two 4's and and 5, for example [2:4][3:3][4:5]. Some descriptions, for example the Wikipedia article listed below, consider this new combination to be standard.
- Min Fanxin (闵凡信) proposes an extra combination "Three Threes or Fours and a Multiple" (三同三四倍), which is set of three 3's or three 4's where the total of the other three ends is a multiple (including multiplication by one) of 3 or 4 respectively. Examples are: [3:3][1:3][1:4] (three 3's and 1+1+4 is 2×3) or [4:4][1:4][1:2] (three 4's and 1+1+2 is 1×4) or [4:4][1:4][6:5] (three 4's and 1+6+5 is 3×4).
Chinese wikipedia entry: https://zh.wikipedia.org/wiki/參禪_(骨牌)
Blog post about the variant 'Generals in mutual harmony': http://blog.sina.com.cn/s/blog_12ae3132b0102y0d2.html.
Blog post about the ShangHai variant with ten sets and no pair: http://blog.sina.com.cn/s/blog_62a3d9fe0100fa3n.html