Thomas Blok has submitted two 𝓟-positions containing 16,24.
2022-04-24.
Thomas Blok has submitted two 𝓟-positions containing 18.
2022-04-21.
Thomas Blok has submitted four new 𝓟-positions with g=2.
2022-04-21.
I have posted Thomas Blok's new article,
Sylver Coinage Even Positions in 14.
It completely analyzes all even positions containing 14.
2022-04-19.
Thomas Blok has submitted two new 𝓟-positions, {24,26,28,32,36,40} (g=2), and {21,30,33,39,45,48} (g=3).
2022-04-14.
Thomas Blok has submitted five new 𝓟-positions with 18.
2022-02-08.
Thomas Blok has submitted a new long 𝓟-position, {24,28,32,34,36,38,40,42,44,46,50,54}. He observes that in many long 𝓟-positions with g=2, a wide range of odd moves have a common odd winning reply. For example, in this position, 409 wins against any odd move from 417 to 439 inclusive, except of course 433 and 437, which 409 eliminates.
2022-02-05.
Thomas Blok has submitted five more 𝓟-positions, including the long position {24,28,30,34,36,38,40,42,44,46,50}.
2021-12-20.
Thomas Blok has contributed ten more 𝓟-positions starting with 16, 18, 20, or 28.
2021-11-23.
Thomas Blok has pointed out that {18,24,27} is 𝓟.
2021-11-22.
Thomas Blok reports three 𝓟-positions with g=3: {18,24,27,30,33}, {18,24,30,33,39}, and {24,27,30,33,36,39,42}.
2021-11-13.
Thomas Blok has contributed five more 𝓟-positions starting with 22.
2021-11-13.
Thomas Blok has contributed three new 𝓟-positions starting with 22.
2021-11-12.
Thomas Blok has contributed four new 𝓟-positions starting with 24.
2021-10-29.
Thomas Blok reports that {24,26,28,30,32,34} is 𝓟. He has also found a new long 𝓟-position, {24,26,28,30,32,34,36,38,40,42,44,46}!
2021-10-21.
Thomas Blok has found an erratum in my list of 𝓟-positions. {18,24,28,30,34,40,50} should be {18,24,28,30,40,50}. I have corrected the page. Thomas has also submitted two new 𝓟-positions, {22,24,26,32,34,42} and {22,24,26,34,36,38,40,42}.
2021-10-07.
Thomas Blok has established that {14,26,64,76} is 𝓟.
2021-10-07.
I have posted a few more 𝓟-positions from Thomas Blok, notably {16,20,46}.
2021-10-04.
Thomas Blok has discovered a new long 𝓟-position! It is {20,26,28,30,32,34,36,38,42}. It has a period of only 120 starting at 347.
2021-10-04.
Thomas Blok has contributed six new 𝓟-positions with 18.
2021-09-22
Thomas Blok has contributed four new 𝓟-positions with 16 or 18.
2021-08-29
Thomas Blok has completed an analysis of all even positions that contain an even number from 2 to 10! You can download it from my Sylver Coinage Page.
2021-06-10
Thomas Blok has contributed four new 𝓟-positions starting with {20,22,24 … }.
2021-05-03
Thomas Blok has found a new 𝓟-position: {10,46,64,82}. While helping him solve long positions, I found an unusually high winning move: {10,46,64,72} [71485]. The only known higher winning moves in positions with g=2 are {6,44,82} [4,5993171] and {6,50,88} [4,197515].
2021-02-17
Thomas Blok has identified more 𝓟-positions with g=2: {16,22,34,36,42}, {16,22,42,50,52}, and {16,22,42,68}.
2021-02-15
Thomas Blok has identified more 𝓟-positions with g=2: {14,20,36,44}, {16,24,30,38,44,52,58}, and {16,24,30,38,52,66}.
2021-02-10
Thomas Blok has identified more 𝓟-positions with g=2: {14,26,44,60}, {14,26,48,50}, {16,22,26,28,34,36,40}, {16,22,26,30,36,40}, {16,22,30,34,40}.
2021-01-15
Thomas Blok has identified more 𝓟-positions with g=2: {14,20,46,52}, {14,22,24,34}, {14,22,26,30,34,38}, {14,22,34,54}, {14,22,30,34,38,46}, {14,24,26,36,46}, {14,24,32,34,36,44}, {14,26,30,38,46,50}, {14,26,32,38,44,50}, and {14,26,36,44}.
2021-01-14
Thomas Blok has identified more 𝓟-positions with g=2: {10,32,54,76}, {10,38,42,46}, {10,38,46,54}, {10,42,44,46}, {14,18,22,26,30,34}, {14,20,22,26,30,32}, and {14,20,24,30,36}.
2021-01-07
Thomas Blok has identified two more 𝓟-positions with g=2: {10,28,36,44} and {10,34,46,58}.
2021-01-05
Thomas Blok has contributed five new 𝓟-positions with g=2: {16,18,22,28,30}, {16,18,24,26,28,30}, {20,22,24,26,28,30,32,34,36}, {22,24,26,28,30,32,34,36,38,40}, and {22,24,26,28,32,42}.
2020-12-23
Thomas Blok has sent me two tables extending the results for {6,…,12} to include 13 and 14.
2020-12-17
Thomas Blok has pointed out that since 14 wins in {10}, [5,14,26] is a complete answer to {10}.
2020-10-31
Jackson Clarke has pointed out that {8} [12,14 is complete! Other even moves that do not reduce to {8,12} or {8,14} have known solutions.
2020-04-09
John Francis, who supplied many of the winning moves in Table 5 of Winning Ways, has written to me to say that the table entry for {9,10,12} is wrong. It is printed as [11,13,14]; it should be [11,13,14,17]. I have confirmed this. The error may have been Francis's, or it may have been the authors'. I have added the table, with Francis's correction and some updates, to this site.
2008-01-25
The short position {18,24,34} is 𝓟.
2008-01-20
The short position {12,32,62} is 𝓟.
2008-01-20
The short position {12,32,58} is 𝓟.
2008-01-20
The short position {12,32,54} is 𝓟.
2002-05-29
The short position {12,28,58} is 𝓟. The lowest possible winning move in {12} with g=2 is 62.
2002-04-29
The long position {20,22,24,26,28,36} is 𝓟. At 797637 it enters a period of 230976. Maybe long 𝓟-positions with g=2 are more common than I thought.
2001-11-07
The long position {16,22,24,26,28,30,34} is 𝓟. See this page for details. It is rare for long positions to be 𝓟.
2001-07-12
The position {12,40,46} is 𝓟. Since {12,40,50} is already known to be 𝓟, the lowest possible winning move in {12} with g=2 is 58.
2001-05-04
The position {18,30,32} is indeed 𝓟. For details see the list of responses.
2001-02-28
I have reorganized the Enders Page.
2001-02-06
The position {18,30,32} is probably 𝓟. I do not think I shall find a winning move in {18}.
2001-01-13
I had hoped to prove that {18,22} is 𝓟, but 79 wins. Meanwhile I have found that 10 is the only winning move in {16,24}.
2000-12-22
The 6-position {6,50,94} has no odd winning move less than 10 to the 8th power. I may abandon this line of inquiry.
2000-12-18
I just added a new 6-position, and it's a whopper: {6,44,82} [4,5993171].
2000-12-18
I have added a table of winning odd moves in even 6-positions.
2000-12-06
I have added some new material to the Enders Page.
2000-11-12
I have started writing a page on enders.
2000-08-25
The position {16,26,88} is probably 𝓟. I had hoped that {16,26} would be 𝓟, because all other derived short positions are 𝓝.
2000-08-23
I have posted a statement of the Progression Hypothesis, which characterizes the ender-status of positions whose moves are in arithmetic progression.
2000-08-03
The position {14,26} is 𝓟. For details see the list of responses.
2000-07-29
The position {14,26} is Probably 𝓟. Of course, even when such a big fish is hooked, it takes a lot of work to land it! This would give a complete first-order solution of {14}: [7, 8, 10, 26].
2000-07-19
The position {12,40,46} is 𝓟. This means that the least possible winning move in {12} with g=2 is 50. Lower moves of the form 4n+2 are answered: {10,12} [7,18]; {12,14} [16]; {12,18} [10,15,16,21]; {12,22} [16,30]; {12,26} [28]; {12,30} [22,...]; {12,34} [20,...]; {12,38} [20,...]; {12,42} [20,...]; {12,46} [40,...].
In fact, any given move in {12} with g=2 is likely to lose because it has many replies that produce short positions. If there exists a winning move with g=2, it is almost certainly too high to compute.