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 4*n*+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.

Back to Sylver Coinage.

Col. George Sicherman [ HOME | MAIL ]