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MASTERMIND STRATEGIES
In the classical Mastermind game where the secret code is a 4 digit repeatable combination of 6 different colours, there are in fact just 5 unique types of first guesses to the game. And a player’s guess will be scored according to its correct (colour and position) and semi-correct (correct colour but wrong position) digits. If we multiply correct number of digits by 10 and add to that number semi-correct digits in the guess, we can show each score by a 2-digit number: 00, 01, 02, ... , 30 and 40. Note that 31 is not a valid score. The table below shows the number of remaining valid possibilities after the first guess for each of the possible combinations and for each possible score:
|
|
1111 |
1112 |
1122 |
1123 |
1234 |
|
00 |
625 |
256 |
256 |
81 |
16 |
|
01 |
0 |
308 |
256 |
276 |
152 |
|
02 |
0 |
61 |
96 |
222 |
312 |
|
03 |
0 |
0 |
16 |
44 |
136 |
|
04 |
0 |
0 |
1 |
2 |
9 |
|
10 |
500 |
317 |
256 |
182 |
108 |
|
11 |
0 |
156 |
208 |
230 |
252 |
|
12 |
0 |
27 |
36 |
84 |
132 |
|
13 |
0 |
0 |
0 |
4 |
8 |
|
20 |
150 |
123 |
114 |
105 |
96 |
|
21 |
0 |
24 |
32 |
40 |
48 |
|
22 |
0 |
3 |
4 |
5 |
6 |
|
30 |
20 |
20 |
20 |
20 |
20 |
|
40 |
1 |
1 |
1 |
1 |
1 |
|
Total: |
1296 |
1296 |
1296 |
1296 |
1296 |
|
Simple |
1 |
2 |
8 |
9 |
52 |
|
Worst Case |
625 |
317 |
256 |
276 |
312 |
|
Entropy |
1.5 |
2.7 |
2.9 |
3.0 |
3.1 |
|
Expected Size |
512.0 |
235.9 |
204.5 |
185.3 |
188.2 |
|
Most Parts |
5 |
11 |
13 |
14 |
14 |
From this table, you can see which game strategy picks which first guess and why it does so by looking at the highlighted cells (these are the first guesses of each classical Mastermind strategy). This table below shows the game solution tree averages for all the classical strategies as well as the Optimal Strategy.
| Strategy | Total Guesses | Average | Maximum Guesses | 6 or more Guesses | Author | Year | |
| Optimal (Max: 6) | 5625 | 4.34028 | 6 | 1 | Lai | 1993 | |
| Optimal (Max: 5) | 5626 | 4.34105 | 5 | 0 | Lai | 1993 | |
| Weighted Entropy | 5646 | 4.35648 | 6 | 3 | Gur | 2021 | |
| Most parts | 5668 | 4.37346 | 6 | 7 | Kooi | 2005 | |
| Expected Size | 5696 | 4.39506 | 6 | 3 | Irving | 1979 | |
| Entropy | 5722 | 4.41512 | 6 | 12 | Neuwirth | 1982 | |
| Worst Case (Min-Max) | 5801 | 4.47608 | 5 | 0 | Knuth | 1977 | |
| Simple | 7471 | 5.76466 | 9 | 853 | Shapiro | 1983 |
