Difference between revisions of "Open problems"
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(Moved seventh row template problem to "formerly open" problems.) |
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* Are there cells other than a1 and b1 which are theoretically losing first moves? | * Are there cells other than a1 and b1 which are theoretically losing first moves? | ||
| − | * Is it true that for every cell (defined in terms of direction and distance from an [[acute corner]]) there is an ''n'' such that for any [[Board]] of size at least ''n'' that cell is a losing [[opening move]]? | + | * Is it true that for every cell (defined in terms of direction and distance from an [[Board#Corners|acute corner]]) there is an ''n'' such that for any [[Board]] of size at least ''n'' that cell is a losing [[opening|opening move]]? |
* Conversely, is it true that, for example, c3 is a winning first move on every Hex board of size at least 5? | * Conversely, is it true that, for example, c3 is a winning first move on every Hex board of size at least 5? | ||
| − | * [[ | + | * Is the [[center opening|center hex]] on every Hex board of [[Board_size|odd size]] a winning opening move? |
| − | * | + | * On boards of all [[board size|sizes]], is every opening move on the [[Board#Diagonals|short diagonal]] winning? |
| − | * | + | * Is the following true? Assume one player is in a winning position (will win with [[optimal play]]) and the opponent plays in a hex X. Let the set A consist of all empty hexes that are members of any path between opponents edges that uses the stone at X. If A is non-empty, A contains a winning move. Otherwise any move is winning, even [[passing]] the turn. (This problem was posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].) |
| + | |||
| + | |||
| + | == Formerly open problems == | ||
| + | |||
| + | === [[Sixth row template problem]] === | ||
| + | |||
| + | Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the sixth row? | ||
| + | |||
| + | '''Answer:''' Yes, [[edge template VI1a]] is such a template. | ||
| + | |||
| + | === Triangle template problem === | ||
| + | |||
| + | Are the templates below valid in their generalization to larger sizes? (This problem was posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].) | ||
| + | |||
| + | <hexboard size="1x1" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | contents="R a1"/><hexboard size="2x2" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(b1,a2,b2)" | ||
| + | contents="R b1"/><hexboard size="3x3" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(c1,a3,c3)" | ||
| + | contents="R c1 a3"/><hexboard size="4x4" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(d1,a4,d4)" | ||
| + | contents="R d1 b3"/><hexboard size="5x5" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(e1,a5,e5)" | ||
| + | contents="R e1 c3 a5"/><hexboard size="6x6" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(f1,a6,f6)" | ||
| + | contents="R f1 d3 b5"/> | ||
| + | |||
| + | '''Answer:''' No. The first one in the sequence that is not connected is the one of height 8. | ||
| + | |||
| + | In fact, using a variant of [[Tom's move]], it is easy to see that even the following triangle, which has more red stones, is not an edge template: | ||
| + | <hexboard size="8x8" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(h1,a8,h8)" | ||
| + | contents="R h1 f3 d5 b7 c7,a8--e8"/> | ||
| + | |||
| + | To see why, imagine that the right edge is a blue edge and that all cells outside the carrier are occupied by Blue. Note that Blue gets a 2nd-and-4th row [[parallel ladder]]. Blue wins by playing the [[Tom's_move#Tall_variant|tall variant of Tom's move]]: | ||
| + | <hexboard size="8x8" | ||
| + | float="inline" | ||
| + | edges="bottom right" | ||
| + | coords="none" | ||
| + | visible="area(h1,a8,h8) g1--a7" | ||
| + | contents="R h1 f3 d5 b7 c7,a8--e8 B g1--a7 B 1:g2 R 2:h2 B 3:f5 R 4:g4 B 5:g3 R 6:h3 B 7:d6 R 8:g5 B 9:f7"/> | ||
| + | |||
| + | There is in fact a template of height 8 continuing the above sequence, but it requires slightly more space: | ||
| + | |||
| + | <hexboard size="8x9" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(h1,a8,i8,i6)" | ||
| + | contents="R h1 f3 d5 b7"/> | ||
| + | The corresponding template of height 9 requires this much space: | ||
| + | |||
| + | <hexboard size="9x11" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(i1,a9,k9,k7,i5)" | ||
| + | contents="R i1 g3 e5 c7 a9"/> | ||
| + | |||
| + | === Seventh row template problem === | ||
| + | |||
| + | Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the seventh row? | ||
| + | |||
| + | '''Answer:''' Yes. See [[Seventh row edge templates]]. | ||
[[category: Open problems]] | [[category: Open problems]] | ||
| + | [[category: Forums]] | ||
Latest revision as of 02:14, 9 November 2023
- Are there cells other than a1 and b1 which are theoretically losing first moves?
- Is it true that for every cell (defined in terms of direction and distance from an acute corner) there is an n such that for any Board of size at least n that cell is a losing opening move?
- Conversely, is it true that, for example, c3 is a winning first move on every Hex board of size at least 5?
- Is the center hex on every Hex board of odd size a winning opening move?
- On boards of all sizes, is every opening move on the short diagonal winning?
- Is the following true? Assume one player is in a winning position (will win with optimal play) and the opponent plays in a hex X. Let the set A consist of all empty hexes that are members of any path between opponents edges that uses the stone at X. If A is non-empty, A contains a winning move. Otherwise any move is winning, even passing the turn. (This problem was posed by Jory in the Little Golem forum.)
Contents
Formerly open problems
Sixth row template problem
Does there exist an edge template which guarantees a secure connection for a piece on the sixth row?
Answer: Yes, edge template VI1a is such a template.
Triangle template problem
Are the templates below valid in their generalization to larger sizes? (This problem was posed by Jory in the Little Golem forum.)
Answer: No. The first one in the sequence that is not connected is the one of height 8.
In fact, using a variant of Tom's move, it is easy to see that even the following triangle, which has more red stones, is not an edge template:
To see why, imagine that the right edge is a blue edge and that all cells outside the carrier are occupied by Blue. Note that Blue gets a 2nd-and-4th row parallel ladder. Blue wins by playing the tall variant of Tom's move:
There is in fact a template of height 8 continuing the above sequence, but it requires slightly more space:
The corresponding template of height 9 requires this much space:
Seventh row template problem
Does there exist an edge template which guarantees a secure connection for a piece on the seventh row?
Answer: Yes. See Seventh row edge templates.
