The Best Initial Guesses

All Mastermind algorithms are sensitive to the first guess chosen. Guessing 1111 first gives far less information about the secret than 1234. There are a limited set of different choices for every combination of pin and color counts.

I've run all games for each deterministic algorithm with each possible initial guess. Raw results are in the .json files in this repo. Tables below give what most people want: the one that yields the lowest maximum turns, and then best average turns.

Patterns of initial guesses

Using different digits for the same pattern isn't useful, nor are shuffled patterns. For example, for a $4p6c$ game these are the unique initial guesses:

$$[1111, 1112, 1122, 1123, 1234]$$

Repetitions of the same pattern, such as 2222, 2111, 3456, 1223, etc. aren't useful as they yield the same information. It should be easy to see that for a first guess, all codewords with the same color (i.e., 1111, 2222, 3333, etc.) give exactly the same amount of information.

Sure, for some secrets one might give a lot more information than another. For instance, if the secret is 2222 then certainly guessing 2222 first is a big win. But over all games, with all digits equally likely, nothing is gained from picking variations on the same pattern.

The best initial guesses for each algorithm

Each table shows the first guess which gives the lowest maximum turns, and then best average turns, over all games. There are some guesses which yield a better average at the expense of a worse maximum; these entries are marked with an asterisk(*). This can be found in the relevant JSON files in this and the results directories; it wasn't worth the clutter in the tables to list both.

Knuth

	2c	3c	4c	5c	6c	7c	8c	9c	10c	11c	12c	13c	14c	15c
2p	12	12	12	12	12	12	12	12	12	12	12	12	12	12
3p	112	112	112	112	123	123	123	123	123	123	123	123	123	123
4p	1112	1123	1123	1123	1122	1234	1234	1234	1234	1234	1234	1234	1234	1123*
5p	11122	11122	11223	11223	11223	11223	11223	11234	12345	12345	12345	12345	12345	11234
6p	111222	111223	112233	112233	111234	112234	112234	112234	112345	112234	112345	123456	123456	112345
7p	1111222	1112233	1112233	1112234	1122334	1122334	1122345	1112234*	1122345	1122334*	1122345	1123456
8p	11111222	11122233	11122334	11122334	11122334	11223344	11223344	11223345

Most Parts

	2c	3c	4c	5c	6c	7c	8c	9c	10c	11c	12c	13c	14c	15c
2p	12	12	12	12	12	12	12	12	12	12	12	12	12	12
3p	112	112	112	123	123	123	123	123	123	123	123	123	123	123
4p	1112	1123	1123	1123	1123	1123	1234	1234	1234	1122*	1234	1234	1234	1234
5p	11122	11123	11223	11223	11122*	11223	11223	11234	11234	11234	11234	12345	12345	12345
6p	111122	111223	111223	111223	112233	112234*	112233	112234	112234	112234	112234	112233*	112345	111234*
7p	1111222	1111223	1112223	1112233	1112233	1234567*	1122334	1122334	1122334	1122334	1122345*	1122334
8p	11112222	11112223	11122233	11122233	11122233	11122334	11122334	11122334

Expected Size

	2c	3c	4c	5c	6c	7c	8c	9c	10c	11c	12c	13c	14c	15c
2p	12	12	12	12	12	12	12	12	12	12	12	12	12	12
3p	112	112	112	112	123	123	123	123	123	123	123	123	123	123
4p	1112	1123	1123	1123	1122*	1234	1234	1234	1123*	1234	1234	1234	1234	1234
5p	11122	11123	11223	11223	11223	11223	11223	11234	11234	11223	11234	12345	11234	11234
6p	111122	111223	111234	111234*	112233	112234	112234	112233	123456	123456	112345*	112345*	112345*	112345*
7p	1111222	1112223	1112233	1112233	1112233	1112233	1234567*	1234567	1234567	1234567
8p	11111222	11122233	11122233	11122334	11112223	11234567	12345678

Entropy

	2c	3c	4c	5c	6c	7c	8c	9c	10c	11c	12c	13c	14c	15c
2p	12	12	12	12	12	12	12	12	12	12	12	12	12	12
3p	112	112	112	123	123	123	123	123	123	123	123	123	123	123
4p	1112	1123	1123	1123	1123	1123	1234	1234	1234	1234	1234	1234	1234	1234
5p	11122	11123	11223	11223	11122*	11223	11223	11234	11234	11234	11234	12345	12345	11223*
6p	111122	111223	111223	112234*	111223	112234	112233	112234	112234	112234	112234	112345	112345	112345
7p	1111222	1112223	1112233	1112233	1112233	1122334	1122334	1122334	1122334	1122345
8p	11112222	11122233	11122233	11122334	11122334	11122334	12345678*

Unique initial guess patterns for various game sizes:

The sets remain constant once $c = p$.

2p2c: 11, 12

3p2c: 111, 112
3p3c: 111, 112, 123

4p2c: 1111, 1112
4p3c: 1111, 1112, 1122, 1123
4p4c: 1111, 1112, 1122, 1123, 1234

5p2c: 11111, 11112
5p3c: 11111, 11112, 11122, 11123, 11223
5p4c: 11111, 11112, 11122, 11123, 11223, 11234
5p5c: 11111, 11112, 11122, 11123, 11223, 11234, 12345

6p2c: 111111, 111112, 111122, 111222
6p3c: 111111, 111112, 111122, 111123, 111222, 111223, 112233
6p4c: 111111, 111112, 111122, 111123, 111222, 111223, 111234, 112233, 112234
6p5c: 111111, 111112, 111122, 111123, 111222, 111223, 111234, 112233, 112234, 112345
6p6c: 111111, 111112, 111122, 111123, 111222, 111223, 111234, 112233, 112234, 112345, 123456

7p2c: 1111111, 1111112, 1111122, 1111222  
7p3c: 1111111, 1111112, 1111122, 1111123, 1111222, 1111223, 1112223, 1112233
7p4c: 1111111, 1111112, 1111122, 1111123, 1111222, 1111223, 1111234, 1112223, 1112233, 1112234, 1122334
7p5c: 1111111, 1111112, 1111122, 1111123, 1111222, 1111223, 1111234, 1112223, 1112233, 1112234, 1112345, 1122334, 1122345
7p6c: 1111111, 1111112, 1111122, 1111123, 1111222, 1111223, 1111234, 1112223, 1112233, 1112234, 1112345, 1122334, 1122345, 1123456
7p7c: 1111111, 1111112, 1111122, 1111123, 1111222, 1111223, 1111234, 1112223, 1112233, 1112234, 1112345, 1122334, 1122345, 1123456, 1234567
 
8p2c: 11111111, 11111112, 11111122, 11111222, 11112222
8p3c: 11111111, 11111112, 11111122, 11111123, 11111222, 11111223, 11112222, 11112223, 11112233, 11122233
8p4c: 11111111, 11111112, 11111122, 11111123, 11111222, 11111223, 11111234, 11112222, 11112223, 11112233, 11112234, 11122233, 11122234, 11122334, 11223344
8p5c: 11111111, 11111112, 11111122, 11111123, 11111222, 11111223, 11111234, 11112222, 11112223, 11112233, 11112234, 11112345, 11122233, 11122234, 11122334, 11122345, 11223344, 11223345
8p6c: 11111111, 11111112, 11111122, 11111123, 11111222, 11111223, 11111234, 11112222, 11112223, 11112233, 11112234, 11112345, 11122233, 11122234, 11122334, 11122345, 11123456, 11223344, 11223345, 11223456
8p7c: 11111111, 11111112, 11111122, 11111123, 11111222, 11111223, 11111234, 11112222, 11112223, 11112233, 11112234, 11112345, 11122233, 11122234, 11122334, 11122345, 11123456, 11223344, 11223345, 11223456, 11234567
8p8c: 11111111, 11111112, 11111122, 11111123, 11111222, 11111223, 11111234, 11112222, 11112223, 11112233, 11112234, 11112345, 11122233, 11122234, 11122334, 11122345, 11123456, 11223344, 11223345, 11223456, 11234567, 12345678