Comment: Quotation from “Green Book”, p.3.5.5.1.(1) “A complete set of tiles is comprised of 6 types of 42 patterns total (Character, Dots, Bams, Winds, Dragons, and Flowers)”.
Questions
Please, provide maximal scoring Mrs. Hudson’s hand under assumptions:
- D = , so the hand looks like AAAABBBBCCC,
- D = , so the hand looks like AAAABBBBCCC,
- C = , D = , so the hand looks like AAAABBBB,
- D = , so the hand looks like AAAABBBBCCC,
- D = , so the hand looks like AAAABBBBCCC,
- D = , so the hand looks like AAAABBBBCCC.
Scoring guidelines:
- correct answer for 1 or 2 questions (of total 6) will be scored for 1 SHP (),
- correct answer for 3 or 4 questions(of total 6) will be scored for 2 SHP (),
- correct answer for 5 or 6 questions (of total 6) will be scored for 3 SHP ().
Solution
This story is about so-called “Pattern Groups”. One cannot find this notion written explicitly in “Green Book” although one can observe it in fan definitions and scoring. Pattern group is such a body of tiles so that a hand of any legal structure (33332, 2222222 etc.) consisting solely of that body will be scored points according to the fan named by that group property. Let’s look at table listing all possible pattern groups:
# |
Pattern Group |
Patterns in Group |
Fan |
Fan Score, pts. |
1 |
Green |
6 |
All Green |
88 |
2 |
Honors |
7 |
All Honors |
64 |
3 |
Terminals |
6 |
All Terminals |
64 |
4C |
Terminals or Honors |
13 |
All Terminals or Honors |
32 |
5 |
Suit Pure |
9 (*3) |
Full Flush |
24 |
6 |
Lower Tiles |
9 |
Lower Tiles |
24 |
7 |
Middle Tiles |
9 |
Middle Tiles |
24 |
8 |
Upper Tiles |
9 |
Upper Tiles |
24 |
9 |
Lower Four |
12 |
Lower Four |
12 |
10 |
Upper Four |
12 |
Upper Four |
12 |
11 |
Reversible |
14 |
Reversible |
8 |
12C |
Suit Mixed |
13+9 (*3) |
Half Flush |
6 |
13 |
Simples |
21 |
All Simples |
2 |
Here “Patterns in Groups” is number of different tile patterns within that group. Pattern group “Suit” has three possible tile implementations – Characters, Bamboos and Dots. “4C” and “12C” are pattern groups combining two other subgroups, each subgroup has at least 2 tiles for pattern group not being scored higher.
Please, note that listed fans score points regardless hand structure. Though, there are several fans which stipulate structure, for instance, “All Even”.
Most striking fact is that any tile pattern can be in several pattern groups. For instance, is in groups: “Green “, “Suit”, “Lower Tiles”, “Lower Four”, “Reversible”, “Simples” (and “Even”). When a hand consists of tiles entering several pattern groups all possible fans based on those pattern groups as-a-whole are considered for summation (omitting incombinable like, for instance, “Lower Tiles” and “Lower Four”).
Now, let’s go back to Mrs. Hudson’s AAAABBBBCCCDDD hand. Points comes from two sources: hand partitioning into sets and pattern groups.
Hand structure
Four identical tiles in a distribution 4-4-3-3 cannot form kong or two pairs (as in Seven Pairs), so, at least one tile of these four should form chow. If one A and one B are in different chows then we have (A is interchangeable with B, and C with D):
Variant 1: (ACD)+(BCD)+(AAABBBCD) or (ACD)*2+(BCD)+(BBB)+(AA). Patterns C and D can make two different chows only if they are adjacent numbers in one suit, so numbers in all patterns go by order A-C-D-B or A-D-C-B in one suit. Current variant costs not a lot: 24 (Full Flush) + 2 (Tile Hog) + 1 (Double Chow) + points for Pung and pattern Groups.
Otherwise, A and B are in the same chow: (ABC)+(AAABBBCCDDD). There exist three ways to partition the second part:
Variant 2. (ABC)+(AAA)+(BBB)+(DDD)+(CC) – Three Pungs, the only restriction is for patterns A, B, C to form Chow,
Variant 3. (ABC)+(ABD)*3+(CC) – Triple Chow, all used patterns are in one suit numbered in order C-A-B-D or D-A-B-C,
Variant 4. (ABC)+(ABD)+(ACD)*2+(BB) – impossible chows, does not work.
As a summary, variants 1 and 3 strictly require one-suited chain of numbered pattern (the difference is what’s inside chain – A/B or C/D), while in variant 2 pung DDD may be any – honor, same or different suit. Variant 2 is more promising both in terms of points and tile patterns combining possibilities.
Pattern Groups
To get maximum from Mrs. Hudson’s hand we need to find such patterns which fit as many high-scored pattern groups as possible.
Answer 1: D = , hand is AAAABBBBCCC. Variant 2 suggests to use --, -- or -- plus one more pattern. Bingo! Since , , are Green tiles then maximum points hand is ( may be replaced by ):
+ + + + .
- All Green = 88 pts.,
- Full Flush = 24 pts.,
- Three Concealed Pungs = 16 pts.,
- Fully Concealed Hand = 4 pts.,
- Tile Hog * 2= 4 pts.
- All Simples = 2 pts.,
Totaling hand value of 138 pts.
Answer 2: D = , hand is AAAABBBBCCC. Variant 2 suggests to use --, -- or -- plus one more pattern. In case this last pattern continues one-suit chain 24-pts. fan can be generated – “Pure Shifted Pungs”, also adds “Reversible Tiles” in comparison with . So, maximum points hand is:
+ + + + .
- Full Flush = 24 pts.,
- Pure Shifted Pungs = 24 pts.,
- Three Concealed Pungs = 16 pts.,
- Reversible Tiles = 8 pts.,
- Fully Concealed Hand = 4 pts.,
- Tile Hog * 2= 4 pts.
- All Simples = 2 pts.,
Totaling hand value of 82 pts.
Answer 3: C = , D = , so the hand looks like AAAABBBB. Variant 2 suggests to use 6 subvariants (combining patterns from Answers 1 and 2): ---, ---, ---, ---, --- or ---. Two subvariants consist of “Reversible Tiles” and only one belongs to “Middle Tiles”! So, maximum points hand is:
+ + + + .
- Middle Tiles = 24 pts.,
- Three Concealed Pungs = 16 pts.,
- Reversible Tiles = 8 pts.,
- Fully Concealed Hand = 4 pts.,
- Tile Hog * 2= 4 pts.
- Double Pung = 2 pts.,
Totaling hand value of 58 pts.
Answers 4-6. What is changed in our analysis when D is a Dragon tile? Are three Dragons the same, in some sense “brothers”? No! They are “stepbrothers” since is Green tile, is Reversible and has no any additional properties. The only possible hand structure is variant 2, additional points can be found in suit pungs. Three hands are:
4. + + + + .
Totaling hand value is 16+6+4+2*2+2+1 = 33 pts.
Current hand can have 6 versions: any of three suits and tiles 1-2-3 instead of 7-8-9.
5. + + + + .
Totaling hand value is 16+8+6+4+2*2+2+1 = 41 pts.
6. + + + + .
Totaling hand value is 88+16+6+4+2*2+2 = 120 pts.
Love it Martin.
Have a lot of fun playing mahjong on t...