Winning Ways has a table showing the winning moves in positions whose elements are subsets of {6,7,8,9,10,11,12}. I reproduce it here with a correction and some additions, shown with green backgrounds.
The even members of the starting position are shown at the left. The odd members are shown at the top. Winning moves are listed in brackets as is customary. An empty pair of brackets indicates that the position is 𝓟. A missing right bracket means that the list of winning moves may be incomplete.
At the foot of this page are two more tables covering positions with 13 and 14, kindly supplied by Thomas Blok.
| {7,9,11} | {7,9} | {7,11} | {7} | {9,11} | {9} | {11} | {} | |
|---|---|---|---|---|---|---|---|---|
| {6,8,10} | [] | [11] | [9] | [] | [4,5,7] | [5] | [] | [4,7,11] |
| {6,8} | [10] | [] | [] | [9,10,11] | [4,5] | [5,7] | [7,10] | [4] |
| {6,10} | [8] | [] | [15] | [8,9] | [4] | [7] | [8,13] | [4] |
| {6} | [] | [8,10,11] | [8,9] | [16] | [4,7] | [] | [26] | [4,9] |
| {8,10,12} | [4,5,6] | [4] | [13] | [5,6] | [13,14,15] | [23] | [6] | [14] |
| {8,12} | [5] | [6] | [6] | [5] | [] | [11,15] | [9,13] | [] |
| {10,12} | [4] | [4,6] | [16] | [] | [] | [11,13,14,17] | [9,14,15,17] | [7,18] |
| {12} | [6] | [15] | [27] | [10] | [8,10] | [6] | [49] | [8 |
| {8,10} | [4,5,6] | [4] | [] | [5,6,11] | [23] | [13,15] | [6,7] | [22] |
| {8} | [5] | [6] | [6,10] | [5] | [12] | [21] | [23] | [12,14] |
| {10} | [4] | [4,6] | [8] | [12] | [12] | [16] | [24,28,47] | [5,14,26] |
| {} | [6] | [19,24] | [24,34] | [] | [13,30] | [6] | [] | * |
The asterisk represents the starting moves that win. This set contains all prime numbers from 5 and up. It does not contain 1, 2, or 3. It contains no composite number divisible by a prime number from 5 and up. It contains no composite number less than 16.
Many of the positions in the table with g=1 were supplied by John Francis. In April 2020, John wrote me to say that he had rechecked his computations. In the book, the entry for {9,10,12} is [11,13,14]. He found that it should be [11,13,14,17].
Years ago I discovered that {10,12,18}, {8,14}, and {10,26} are 𝓟. I have updated the corresponding entries in the table.
In October 2020 Jackson Clarke pointed out to me that {8} [12,14 is complete. In December 2020 Thomas Blok pointed out to me that since 14! wins in {10}, [5,14,26] is complete for {10}.
| {7,9,11,13} | {7,9,13} | {7,11,13} | {7,13} | {9,11,13} | {9,13} | {11,13} | {13} | |
|---|---|---|---|---|---|---|---|---|
| {6,8,10} | [] | [11] | [9] | [] | [4,5,7] | [5] | [15] | [7] |
| {6,8} | [10] | [] | [] | [9,10,11] | [4,5] | [5,7] | [7] | [17] |
| {6,10} | [8] | [] | [15] | [8,9] | [4] | [7] | [] | [11] |
| {6} | [] | [8,10,11] | [8,9] | [16] | [4,7] | [14,16] | [10,15] | [23,28] |
| {8,10,12} | [4,5,6] | [4] | [] | [5,6,11] | [] | [11,14,15] | [7,9] | [17] |
| {8,12} | [5] | [6] | [4,6,10] | [4,5] | [10,14,15] | [23] | [] | [11,15] |
| {10,12} | [4] | [4,5,6] | [5,8] | [15,16,18] | [8,14,15] | [] | [6] | [9] |
| {12} | [6] | [5] | [4,5] | [4] | [16,17,19] | [10] | [8] | [59] |
| {8,10} | [4,5,6] | [4] | [12] | [5,6] | [12,14,15] | [] | [17] | [9] |
| {8} | [5] | [6] | [4,6] | [4,5] | [23] | [10,14,15] | [12] | [25] |
| {10} | [4] | [4,5,6] | [5] | [19,22] | [25] | [8,12] | [6,15] | [28] |
| {} | [6] | [5] | [4,5] | [4] | [] | [11] | [9,19] | [] |
| {9,11,13} | {9,11} | {9,13} | {9} | {11,13} | {11} | {13} | {} | |
|---|---|---|---|---|---|---|---|---|
| {6,10,14} | [4] | [4] | [7] | [7] | [5] | [8] | [5] | [4,17,25] |
| {6,14} | [4,7] | [4,7] | [] | [13] | [5] | [15] | [5,9] | [4] |
| {8,10,12,14} | [15] | [] | [] | [11,13,15] | [7] | [6,9] | [9,15] | [] |
| {8,10,14} | [] | [12,13,15] | [11,12,15] | [21] | [9] | [6,7] | [17] | [12,19,23] |
| {8,12,14} | [] | [10,13,15] | [10,11] | [19] | [9,15] | [17,18,21] | [31] | [10] |
| {8,14} | [10,12,15] | [21] | [] | [13,20] | [31] | [15] | [9,19] | [] |
| {10,12,14} | [] | [8,13,15] | [8,11,15] | [] | [9,15,16,17] | [] | [31] | [7,8,9,11] |
| {10,14} | [8,12,15] | [26] | [16] | [12,15,17] | [29] | [12,15,16] | [15,32] | [] |
| {12,14} | [8,10] | [16,17,19] | [6,16,17] | [10,25] | [43] | [10,29] | [19] | [16] |
| {14} | [30] | [15] | [6,8] | [22,48] | [17,32] | [93] | [155] | [7,8,10,26] |