User:Hexanna - Revision history

Hexanna: solved 4x4 dark hex

2026-09-10T03:49:07Z

solved 4x4 dark hex

Hexanna: rex+ is a second player win on 5x5

2026-09-05T22:33:45Z

rex+ is a second player win on 5x5

Hexanna: solved 3x4 dark hex!

2026-08-30T00:02:00Z

solved 3x4 dark hex!

Hexanna: 3x4 dark hex improved bounds

2026-08-29T18:56:39Z

3x4 dark hex improved bounds

Hexanna: template VI1d

2026-08-29T14:52:14Z

template VI1d

Hexanna: expand on V1k minimality

2026-08-29T03:02:54Z

expand on V1k minimality

Hexanna: template V1k

2026-08-29T02:10:01Z

template V1k

Hexanna: template V1j

2026-08-28T22:00:53Z

template V1j

Hexanna: templates IV1g, IV1h

2026-08-28T06:26:55Z

templates IV1g, IV1h

Hexanna: V1i is now a template

2026-08-28T03:16:11Z

V1i is now a template

@@ Line 15: / Line 15: @@
 * Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.
-Potential exact solution to 3&times;4 Classic Dark Hex: The first player wins with a probability of exactly 14279/67484 (0.211590895619702448) under optimal play. See https://github.com/hexanna1/dark-hex for a strategy certificate. The previous best appears to be [1/7, 1/4], as stated in [https://www.sfu.ca/~jed/Conferences/CCC2023abstracts.pdf Notes on Dark Hex].
+Solution to 4&times;4 Classic Dark Hex: The first player wins with a probability of exactly 5345119702052171/6736824633547900 (0.793418263470100504) under optimal play.
 Rex+ is a second player win on 5&times;5 (and smaller board sizes). See the [https://hexanna1.github.io/hex-study/rexplus.html website] and [https://github.com/hexanna1/rexplus solver code].

← Older revision		Revision as of 22:33, 5 September 2026
Line 16:		Line 16:

	Potential exact solution to 3×4 Classic Dark Hex: The first player wins with a probability of exactly 14279/67484 (0.211590895619702448) under optimal play. See https://github.com/hexanna1/dark-hex for a strategy certificate. The previous best appears to be [1/7, 1/4], as stated in [https://www.sfu.ca/~jed/Conferences/CCC2023abstracts.pdf Notes on Dark Hex].		Potential exact solution to 3×4 Classic Dark Hex: The first player wins with a probability of exactly 14279/67484 (0.211590895619702448) under optimal play. See https://github.com/hexanna1/dark-hex for a strategy certificate. The previous best appears to be [1/7, 1/4], as stated in [https://www.sfu.ca/~jed/Conferences/CCC2023abstracts.pdf Notes on Dark Hex].
		+
		+	Rex+ is a second player win on 5×5 (and smaller board sizes). See the [https://hexanna1.github.io/hex-study/rexplus.html website] and [https://github.com/hexanna1/rexplus solver code].

	==new templates==		==new templates==

@@ Line 15: / Line 15: @@
 * Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.
-Potential new bounds on 3&times;4 Classic Dark Hex: The first player wins with a probability between 5642141/26723476 (0.211130505627336803) and 5654469/26723476 (0.211591822860169837) under optimal play. See https://github.com/hexanna1/dark-hex for a strategy certificate. The previous best appears to be [1/7, 1/4], as stated in [https://www.sfu.ca/~jed/Conferences/CCC2023abstracts.pdf Notes on Dark Hex].
+Potential exact solution to 3&times;4 Classic Dark Hex: The first player wins with a probability of exactly 14279/67484 (0.211590895619702448) under optimal play. See https://github.com/hexanna1/dark-hex for a strategy certificate. The previous best appears to be [1/7, 1/4], as stated in [https://www.sfu.ca/~jed/Conferences/CCC2023abstracts.pdf Notes on Dark Hex].
 ==new templates==

@@ Line 14: / Line 14: @@
 * [https://hexanna1.github.io/hex-study/openings.html Opening explorer] with opening lines on sizes 11&ndash;14 and 17
 * Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.
 ==new templates==

@@ Line 116: / Line 116: @@
      />
-Validity and minimality checked with Mohex. Minimality took roughly 40 core-hours to prove; validity only took around 0.3 core-hours. Unlike V1i, which had a single deletion that was very difficult to prove, V1k had many moderately difficult ones but no exceptionally difficult one. I used a mix of 1k- and 10k-visit KataHex searches for move ordering only, and that was sufficient to prove each deletion.
+Validity and minimality checked with Mohex. Minimality took roughly 40 CPU-hours to prove; validity only took around 0.3 CPU-hours. Unlike V1i, which had a single deletion that was very difficult to prove, V1k had many moderately difficult ones but no exceptionally difficult one. I used a mix of 1k- and 10k-visit KataHex searches for move ordering only, and that was sufficient to prove each deletion.
 Many lines in this template make use of [[Fifth_row_edge_templates#V-2-m]]. I tried an experiment where I partitioned the template into V2m on the right side and the complement on the left side, as shown below. In minimality proofs, a cell is deleted from the carrier, and Blue has to prove that the template can be defended against each Red response. I added a rule to Mohex to restrict Blue's response to the same half of the board as Red's previous response, at certain positions in the search tree where the cells marked (*) were both red (as they are in V2m). This rule is sound, in that if Blue is still able to defend against each Red response, the deletion is a proven Blue win in the unrestricted setting, but it's not complete (not every deletion can be proven with this rule enabled). Indeed, some deletions, like (-) below, became winning for Red under this restriction, leading to an inconclusive result that still had to be proved the "normal" way. The idea was that many positions were tricky for Mohex because of high branching when a template is effectively split in two, because Blue would play delaying moves in the irrelevant half of the board. This rule saved time on some deletions, but only modestly, and I rechecked each deletion without the rule. A refined version of this approach could be useful for more complicated templates.
@@ Line 126: / Line 126: @@
      contents="R k2 E *:(n2 m3) -:q2 S red:area(n2,m3,k6,t6,t4,r2) h6"
      />
 ===infinite family of templates===

← Older revision		Revision as of 03:02, 29 August 2026
Line 117:		Line 117:

	Validity and minimality checked with Mohex. Minimality took roughly 40 core-hours to prove; validity only took around 0.3 core-hours. Unlike V1i, which had a single deletion that was very difficult to prove, V1k had many moderately difficult ones but no exceptionally difficult one. I used a mix of 1k- and 10k-visit KataHex searches for move ordering only, and that was sufficient to prove each deletion.		Validity and minimality checked with Mohex. Minimality took roughly 40 core-hours to prove; validity only took around 0.3 core-hours. Unlike V1i, which had a single deletion that was very difficult to prove, V1k had many moderately difficult ones but no exceptionally difficult one. I used a mix of 1k- and 10k-visit KataHex searches for move ordering only, and that was sufficient to prove each deletion.
		+
		+	Many lines in this template make use of [[Fifth_row_edge_templates#V-2-m]]. I tried an experiment where I partitioned the template into V2m on the right side and the complement on the left side, as shown below. In minimality proofs, a cell is deleted from the carrier, and Blue has to prove that the template can be defended against each Red response. I added a rule to Mohex to restrict Blue's response to the same half of the board as Red's previous response, at certain positions in the search tree where the cells marked (*) were both red (as they are in V2m). This rule is sound, in that if Blue is still able to defend against each Red response, the deletion is a proven Blue win in the unrestricted setting, but it's not complete (not every deletion can be proven with this rule enabled). Indeed, some deletions, like (-) below, became winning for Red under this restriction, leading to an inconclusive result that still had to be proved the "normal" way. The idea was that many positions were tricky for Mohex because of high branching when a template is effectively split in two, because Blue would play delaying moves in the irrelevant half of the board. This rule saved time on some deletions, but only modestly, and I rechecked each deletion without the rule. A refined version of this approach could be useful for more complicated templates.
		+
		+	<hexboard size="6x20"
		+	coords="hide"
		+	edges="bottom"
		+	visible="area(l1,g2,e3,c4,a6,t6,t4,s3,r2,n1)"
		+	contents="R k2 E *:(n2 m3) -:q2 S red:area(n2,m3,k6,t6,t4,r2) h6"
		+	/>

	===infinite family of templates===		===infinite family of templates===

← Older revision		Revision as of 02:10, 29 August 2026
Line 106:		Line 106:

	Validity and minimality checked with Mohex.		Validity and minimality checked with Mohex.
		+
		+	Fifth row template ('''V1k'''), width 20, carrier size 86 [https://hexworld.org/board/#20c1,k1:pk2:pk3:pk4:pk5:pk6:pk7:pk8:pk9:pk10:pk11:pk12:pk13:pk14:pk15:pk16a19:pb18:pc17:pd17:pe16:pf16:pg15:ph15:ph20:pi15:pj15:pl14:pm14:pn14:po14:po15:pp15:pq15:pr15:ps15:ps16:pt16:pt17:p HexWorld link]
		+
		+	<hexboard size="6x20"
		+	coords="hide"
		+	edges="bottom"
		+	visible="area(l1,g2,e3,c4,a6,t6,t4,s3,r2,n1)"
		+	contents="R k2 S h6"
		+	/>
		+
		+	Validity and minimality checked with Mohex. Minimality took roughly 40 core-hours to prove; validity only took around 0.3 core-hours. Unlike V1i, which had a single deletion that was very difficult to prove, V1k had many moderately difficult ones but no exceptionally difficult one. I used a mix of 1k- and 10k-visit KataHex searches for move ordering only, and that was sufficient to prove each deletion.

	===infinite family of templates===		===infinite family of templates===

← Older revision		Revision as of 22:00, 28 August 2026
Line 95:		Line 95:

	Validity and minimality checked with Mohex. By far, the hardest thing to prove was minimality after deleting (-), because of Blue's response at (). The second hardest thing was proving validity of Blue's response at (). I used a modified version of Mohex/Benzene that uses KataHex only for move ordering in the DFPN search; all proof results are established independently by Mohex. The code and a script to run the full proof (with the hardcoded move order, no KataHex necessary) are at [https://github.com/hexanna1/benzene-vanilla-cmake this repository]. The script takes tens of minutes to hours to run.		Validity and minimality checked with Mohex. By far, the hardest thing to prove was minimality after deleting (-), because of Blue's response at (). The second hardest thing was proving validity of Blue's response at (). I used a modified version of Mohex/Benzene that uses KataHex only for move ordering in the DFPN search; all proof results are established independently by Mohex. The code and a script to run the full proof (with the hardcoded move order, no KataHex necessary) are at [https://github.com/hexanna1/benzene-vanilla-cmake this repository]. The script takes tens of minutes to hours to run.
		+
		+	Fifth row template ('''V1j'''), width 17, carrier size 71 [https://hexworld.org/board/#17c1,h1:ph2:ph3:ph4:ph5:ph6:ph7:ph8:ph9:ph10:ph11:ph12:ph13a16:pb15:pc14:pd14:pe13:pe16:pf13:pg12:pi11:pj11:pk11:pl11:pl12:pm12:pn12:po12:pp12:pp13:pq13:pq14:p HexWorld link]
		+
		+	<hexboard size="6x17"
		+	coords="hide"
		+	edges="bottom"
		+	visible="area(i1,g2,e3,c4,a6,q6,q4,p3,o2,k1)"
		+	contents="R h2 S e5"
		+	/>
		+
		+	Validity and minimality checked with Mohex.

	===infinite family of templates===		===infinite family of templates===

@@ Line 38: / Line 38: @@
 Validity and minimality checked with Mohex.
 Fifth row template ('''V1g''', if we first name [https://www.mathstat.dal.ca/~selinger/templates/svg/template-5-wj.svg V1e] and [https://www.mathstat.dal.ca/~selinger/templates/svg/template-5-wyg.svg V1f]), width 16, carrier size 66 [https://hexworld.org/board/#16c1,g1:pg2:pg3:pg4:pg5:pg6:pg7:pg8:pg9:pg10:pg11:pg12a15:pb14:pc13:pc16:pd13:pe12:pf12:ph10:pi10:pj10:pk10:pk11:pl11:pm11:pn11:po11:po12:pp12:pp13:p HexWorld link]
@@ Line 46: / Line 70: @@
      visible="area(h1,e3,c4,a6,p6,p4,n2,j1)"
      contents="R g2 S c6"
-  />
+    />
 Validity and minimality checked with Mohex.
@@ Line 81: / Line 105: @@
      visible="area(h1,e3,c4,a6,q6,q4,o2,j1)-l2"
      contents="R g2 S c6 E *:l3 *:l4 *:l5 *:l6"
-  />
+    />
 <hexboard size="6x15"
@@ Line 88: / Line 112: @@
      visible="area(f1,a6,o6,o4,m2,h1)-j2"
      contents="R e2 E *:j3 *:j4 *:j5 *:j6"
-  />
+    />
 ==basic 1-2* hex strategy==

@@ Line 61: / Line 61: @@
 Validity and minimality checked with Mohex.
-Fifth row template '''candidate''' ('''V1i'''), width 18, carrier size 77 [https://hexworld.org/board/#18c1,h1:ph2:ph3:ph4:ph5:ph6:ph7:ph8:ph9:ph10:ph11:ph12:ph13:ph14a17:pb16:pc15:pd15:pe14:pf14:pg13:pi12:pj12:pj18:pk12:pl12:pm12:pm13:pn13:po13:pp13:pq13:pq14:pr14:pr15:p HexWorld link]
+Fifth row template ('''V1i'''), width 18, carrier size 77 [https://hexworld.org/board/#18c1,h1:ph2:ph3:ph4:ph5:ph6:ph7:ph8:ph9:ph10:ph11:ph12:ph13:ph14a17:pb16:pc15:pd15:pe14:pf14:pg13:pi12:pj12:pj18:pk12:pl12:pm12:pm13:pn13:po13:pp13:pq13:pq14:pr14:pr15:p HexWorld link]
 <hexboard size="6x18"
@@ Line 67: / Line 67: @@
      edges="bottom"
      visible="area(i1,g2,e3,c4,a6,r6,r4,q3,p2,l1)"
-     contents="R h2 E *:i1 S j6"
+     contents="R h2 E *:g5 -:i1 S j6"
      />
-Validity checked with Mohex, but I'm still working on minimality. The pattern after deleting (*) is difficult to prove losing for Red. More details to come soon.
+Validity and minimality checked with Mohex. By far, the hardest thing to prove was minimality after deleting (-), because of Blue's response at (*). The second hardest thing was proving validity of Blue's response at (*). I used a modified version of Mohex/Benzene that uses KataHex only for move ordering in the DFPN search; all proof results are established independently by Mohex. The code and a script to run the full proof (with the hardcoded move order, no KataHex necessary) are at [https://github.com/hexanna1/benzene-vanilla-cmake this repository]. The script takes tens of minutes to hours to run.
 ===infinite family of templates===