Talk:Wheel - Revision history

Selinger: Added a 7x7 example

2023-04-19T00:26:00Z

Added a 7x7 example

Selinger: Oops, forgot a dead cell.

2023-04-18T21:37:46Z

Oops, forgot a dead cell.

Selinger: Further simplifcation to 8x10.

2023-04-18T21:35:22Z

Further simplifcation to 8x10.

Selinger: Further simplified the position

2023-04-18T17:51:50Z

Further simplified the position

Wurfmaul: 10x10 position

2023-04-18T00:56:44Z

10x10 position

Wurfmaul: 11x11 position

2023-04-17T22:07:12Z

11x11 position

Hexanna at 01:37, 17 April 2023

2023-04-17T01:37:16Z

Hexanna: Can intrusion into the center of a wheel ever be the unique winning move?

2023-04-17T01:09:31Z

Can intrusion into the center of a wheel ever be the unique winning move?

Turing: Copied/adapted page from http://hexwiki.tk

2005-02-10T07:18:09Z

Copied/adapted page from http://hexwiki.tk

New page

<hex>R5 C5 Vc2 Vd3 Vb4 Hc4</hex>

This pattern is not what I would call a Wheel. It is at most a Broken wheel. I think somebody called it a U-turn. — taral
:Yes, granted. My initial feeling was that it was important to point out that this related position is weak, but maybe it's not important enough to merit mention on this page. We could always create a "U-turn" page, or just refer to the example in the tutorial. [[User:Turing|turing]] 08:18, 10 Feb 2005 (CET)

← Older revision		Revision as of 00:26, 19 April 2023
Line 24:		Line 24:
	By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)		By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)
	Using [[Theorems about templates\|corner clipping]], we can reduce the number of empty hexes by 4 more, and reduce the size to 8x10: [https://hexworld.org/board/#8x10,f1a1h2a2d4a3h4a4d5a5h5a6c6a7d6a8h6b1h7b2a9b3h9b4:pb5:pb6:pc1:pc2:pc3:p:p:pc4:pc5:pc7:pd1:pd2:pd3:pd7:pe5:pe8:pf5:pf6:pg5:ph10:pe1 here]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 21:35, 18 April 2023 (UTC)		Using [[Theorems about templates\|corner clipping]], we can reduce the number of empty hexes by 4 more, and reduce the size to 8x10: [https://hexworld.org/board/#8x10,f1a1h2a2d4a3h4a4d5a5h5a6c6a7d6a8h6b1h7b2a9b3h9b4:pb5:pb6:pc1:pc2:pc3:p:p:pc4:pc5:pc7:pd1:pd2:pd3:pd7:pe5:pe8:pf5:pf6:pg5:ph10:pe1 here]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 21:35, 18 April 2023 (UTC)
		+
		+	In fact, the following is a 7x7 example, which also works for the [[Interior_template#The_hammock\|hammock]] and not just the wheel, and uses only 5 empty cells outside of the hammock:
		+	<hexboard size="7x7"
		+	contents="B a1--e1 g1 a2 b2 a3 f3 e4 b5 c5 c6 d6 g6 g7--e7 f5 R f1 d2 g2--g5 e3 a4--a7--d7 b6 c4 d5
		+	E A:(c3 e2 d4) B:d3"
		+	/>
		+	[[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) and [[User:Demer\|Demer]] ([[User talk:Demer\|talk]]) 00:26, 19 April 2023 (UTC)

← Older revision		Revision as of 21:37, 18 April 2023
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	By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)		By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)
−	Using [[Theorems about templates\|corner clipping]], we can reduce the number of empty hexes by 4 more, and reduce the size to 8x10: [https://hexworld.org/board/#8x10,f1a1h2a2d4a3h4a4d5a5h5a6c6a7d6a8h6b1h7b2a9b3h9b4:pb5:pb6:pc1:pc2:pc3:p:p:pc4:pc5:pc7:pd1:pd2:pd3:pd7:pe5:pe8:pf5:pf6:pg5:ph10 here]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 21:35, 18 April 2023 (UTC)	+	Using [[Theorems about templates\|corner clipping]], we can reduce the number of empty hexes by 4 more, and reduce the size to 8x10: [https://hexworld.org/board/#8x10,f1a1h2a2d4a3h4a4d5a5h5a6c6a7d6a8h6b1h7b2a9b3h9b4:pb5:pb6:pc1:pc2:pc3:p:p:pc4:pc5:pc7:pd1:pd2:pd3:pd7:pe5:pe8:pf5:pf6:pg5:ph10:pe1 here]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 21:35, 18 April 2023 (UTC)

← Older revision		Revision as of 21:35, 18 April 2023
Line 23:		Line 23:

	By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)		By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)
		+	Using [[Theorems about templates\|corner clipping]], we can reduce the number of empty hexes by 4 more, and reduce the size to 8x10: [https://hexworld.org/board/#8x10,f1a1h2a2d4a3h4a4d5a5h5a6c6a7d6a8h6b1h7b2a9b3h9b4:pb5:pb6:pc1:pc2:pc3:p:p:pc4:pc5:pc7:pd1:pd2:pd3:pd7:pe5:pe8:pf5:pf6:pg5:ph10 here]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 21:35, 18 April 2023 (UTC)

← Older revision		Revision as of 17:51, 18 April 2023
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	[https://hexworld.org/board/#11n,d7b9e7d8e6e8e5f9i6g7i7f6i8g6i5e4g2j2i3j3j5d4j6c4j7b4k5j9k6j10 Here] is a simplification for which the dfpn solver can prove that B is the only winning move. Surely it can be simplified further. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 22:07, 17 April 2023 (UTC)		[https://hexworld.org/board/#11n,d7b9e7d8e6e8e5f9i6g7i7f6i8g6i5e4g2j2i3j3j5d4j6c4j7b4k5j9k6j10 Here] is a simplification for which the dfpn solver can prove that B is the only winning move. Surely it can be simplified further. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 22:07, 17 April 2023 (UTC)
	Actually the top-most row and right-most column of this position can simply be discarded to reach a [https://hexworld.org/board/#10n,d6b8e6d7e5e7e4g6i7f8i6j8i5j9i4j2i2j1g1e3j4d3j5c3j6b3d5g5d4f5 10x10 position]. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 00:56, 18 April 2023 (UTC)		Actually the top-most row and right-most column of this position can simply be discarded to reach a [https://hexworld.org/board/#10n,d6b8e6d7e5e7e4g6i7f8i6j8i5j9i4j2i2j1g1e3j4d3j5c3j6b3d5g5d4f5 10x10 position]. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 00:56, 18 April 2023 (UTC)
		+
		+	By filling in the [[captured cell\|captured]] areas, Wurfmaul's position can be further simplified to [https://hexworld.org/board/#9x10,g1a1i2a2e4a3i4a4e5a5i5a6d6a7e6a8i6a9i7b1:pb2:pb3:pb4:pb5:pb6:pb7:p:p:pb8:pc1:pc2:pc3:pc4:pc5:pc6:pd1:pd2:pd3:pd4:pd5:pe1:pe2:pe3:pf1:pf5:pg5:ph5:pg6:pd7:pe7:pf8 this]. [[User:Selinger\|Selinger]] ([[User talk:Selinger\|talk]]) 17:51, 18 April 2023 (UTC)

← Older revision		Revision as of 00:56, 18 April 2023
Line 20:		Line 20:

	[https://hexworld.org/board/#11n,d7b9e7d8e6e8e5f9i6g7i7f6i8g6i5e4g2j2i3j3j5d4j6c4j7b4k5j9k6j10 Here] is a simplification for which the dfpn solver can prove that B is the only winning move. Surely it can be simplified further. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 22:07, 17 April 2023 (UTC)		[https://hexworld.org/board/#11n,d7b9e7d8e6e8e5f9i6g7i7f6i8g6i5e4g2j2i3j3j5d4j6c4j7b4k5j9k6j10 Here] is a simplification for which the dfpn solver can prove that B is the only winning move. Surely it can be simplified further. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 22:07, 17 April 2023 (UTC)
		+	Actually the top-most row and right-most column of this position can simply be discarded to reach a [https://hexworld.org/board/#10n,d6b8e6d7e5e7e4g6i7f8i6j8i5j9i4j2i2j1g1e3j4d3j5c3j6b3d5g5d4f5 10x10 position]. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 00:56, 18 April 2023 (UTC)

← Older revision		Revision as of 22:07, 17 April 2023
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	It would be nice if someone could come up with an example small enough that one can ''prove'' the intrusion at B is the unique winning move. I suspect the smallest example might not even fit on a 5x5 or 6x6 board, since it's a fickle matter getting the condition to hold. [[User:Hexanna\|Hexanna]] ([[User talk:Hexanna\|talk]]) 01:09, 17 April 2023 (UTC)		It would be nice if someone could come up with an example small enough that one can ''prove'' the intrusion at B is the unique winning move. I suspect the smallest example might not even fit on a 5x5 or 6x6 board, since it's a fickle matter getting the condition to hold. [[User:Hexanna\|Hexanna]] ([[User talk:Hexanna\|talk]]) 01:09, 17 April 2023 (UTC)
		+
		+	[https://hexworld.org/board/#11n,d7b9e7d8e6e8e5f9i6g7i7f6i8g6i5e4g2j2i3j3j5d4j6c4j7b4k5j9k6j10 Here] is a simplification for which the dfpn solver can prove that B is the only winning move. Surely it can be simplified further. --[[User:Wurfmaul\|Wurfmaul]] ([[User talk:Wurfmaul\|talk]]) 22:07, 17 April 2023 (UTC)

@@ Line 15: / Line 15: @@
 Now, this might not be very interesting because playing at d2 just delays the "real" winning d4 move. A more interesting question: is there a position where playing at B is the ''unique'' winning move? This seems trickier than it looks, in many winning positions there's a useless bridge intrusion that also preserves the win for instance, so the move at B isn't unique. Intuitively, I think that in order for the intrusion at B to be better than all intrusions at A, the intrusion at B needs to serve the function of asking a [[question]].
-"lazyplayer" from LittleGolem showed me [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10 a position] loosely based from [https://littlegolem.net/jsp/game/game.jsp?gid=2359962 this game]. I was surprised when I put this in KataHex, because it was the first position I saw where an intrusion at B (specifically, f10) was winning but intrusions at A were losing. But, f10 might not be unique; KataHex thinks c11 or i5 could also win. I played around a bit to see if I could make f10 unique &mdash; it was tricky to get KataHex to not think c11 was also winning or close to 50%. Eventually, I found [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10i5i2i3j2h3d4 a position] where KataHex thinks, after about 50k visits, that f10 is winning (90.3% win rate), and the second best move has only an 8.6% win rate. So it appears that this last position has a unique winning move at B, but I only have KataHex evaluations and not a solid proof.
+"lazyplayer" from LittleGolem showed me [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10 a position] loosely based from [https://littlegolem.net/jsp/game/game.jsp?gid=2359962 this game]. I was surprised when I put this in KataHex, because it was the first position I saw where an intrusion at B (specifically, f10) was winning but intrusions at A were losing. But, f10 might not be unique; KataHex thinks c11 or i5 could also win. I played around a bit to see if I could make f10 unique &mdash; it was tricky to get KataHex to not think c11 was also winning or close to 50%. Eventually, I found [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10i5i2i3j2h3d4 a position] where KataHex thinks, after about 50k visits, that f10 is winning (90.3% win rate), and the second best move has only an 8.6% win rate. So it appears that this last position has a unique winning move at B, but I only have KataHex evaluations and not a solid proof. (Note for reproducibility: I used the katahex_model_20220618.bin.gz net with a high value of analysisWideRootNoise, 0.5, to reduce blind spots.)
 It would be nice if someone could come up with an example small enough that one can ''prove'' the intrusion at B is the unique winning move. I suspect the smallest example might not even fit on a 5x5 or 6x6 board, since it's a fickle matter getting the condition to hold. [[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 01:09, 17 April 2023 (UTC)

← Older revision		Revision as of 01:09, 17 April 2023
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	This pattern is not what I would call a Wheel. It is at most a Broken wheel. I think somebody called it a U-turn. — taral		This pattern is not what I would call a Wheel. It is at most a Broken wheel. I think somebody called it a U-turn. — taral
	:Yes, granted. My initial feeling was that it was important to point out that this related position is weak, but maybe it's not important enough to merit mention on this page. We could always create a "U-turn" page, or just refer to the example in the tutorial. [[User:Turing\|turing]] 08:18, 10 Feb 2005 (CET)		:Yes, granted. My initial feeling was that it was important to point out that this related position is weak, but maybe it's not important enough to merit mention on this page. We could always create a "U-turn" page, or just refer to the example in the tutorial. [[User:Turing\|turing]] 08:18, 10 Feb 2005 (CET)
		+
		+
		+	''Blue should never attempt to intrude by playing at B.'' I believe this is false. Here's a simple example (Blue to move) where Blue's only winning moves are at one of (*). In particular, d2 is winning for Blue, while all of the hexes marked A are losing.
		+
		+	<hexboard size="5x5"
		+	coords="show"
		+	edges="show"
		+	contents="R d1 e2 c3 e5 B c1 b3 b5 E *:(d2 d4) A:(e1 c2 d3)"
		+	/>
		+
		+	Now, this might not be very interesting because playing at d2 just delays the "real" winning d4 move. A more interesting question: is there a position where playing at B is the ''unique'' winning move? This seems trickier than it looks, in many winning positions there's a useless bridge intrusion that also preserves the win for instance, so the move at B isn't unique. Intuitively, I think that in order for the intrusion at B to be better than all intrusions at A, the intrusion at B needs to serve the function of asking a [[question]].
		+
		+	"lazyplayer" from LittleGolem showed me [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10 a position] loosely based from [https://littlegolem.net/jsp/game/game.jsp?gid=2359962 this game]. I was surprised when I put this in KataHex, because it was the first position I saw where an intrusion at B (specifically, f10) was winning but intrusions at A were losing. But, f10 might not be unique; KataHex thinks c11 or i5 could also win. I played around a bit to see if I could make f10 unique — it was tricky to get KataHex to not think c11 was also winning or close to 50%. Eventually, I found [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10i5i2i3j2h3d4 a position] where KataHex thinks, after about 50k visits, that f10 is winning (90.3% win rate), and the second best move has only an 8.6% win rate. So it appears that this last position has a unique winning move at B, but I only have KataHex evaluations and not a solid proof.
		+
		+	It would be nice if someone could come up with an example small enough that one can ''prove'' the intrusion at B is the unique winning move. I suspect the smallest example might not even fit on a 5x5 or 6x6 board, since it's a fickle matter getting the condition to hold. [[User:Hexanna\|Hexanna]] ([[User talk:Hexanna\|talk]]) 01:09, 17 April 2023 (UTC)