Difference between revisions of "Open problems"
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(Added an old question from LG forum.) |
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* Is the following true? Assume one player is in a winning position (will win with [[optimal play]]) and 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 are 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].) | * Is the following true? Assume one player is in a winning position (will win with [[optimal play]]) and 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 are 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].) | ||
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== Formerly open problems == | == 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? | + | * [[Sixth row template problem]]: Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the sixth row?<br> '''Answer:''' Yes, and [[edge template VI1a]] is such a template. |
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| + | * 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].) <br> <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"/><hexboard size="4x4" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(d1,a4,d4)" | ||
| + | contents="R d1 d3"/><hexboard size="5x5" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(e1,a5,e5)" | ||
| + | contents="R e1 e3 e5"/><hexboard size="6x6" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(f1,a6,f6)" | ||
| + | contents="R f1 f3 f5"/> <br> '''Answer:''' No. The first one in the sequence that is not connected is the one of height 8. Instead, it requires this much space: <br> <hexboard size="8x9" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(i1,d6,c6,a8,i8)" | ||
| + | contents="R i1 i3 i5 i7"/><br>The corresponding template of height 9 requires this much space:<br><hexboard size="9x11" | ||
| + | float="inline" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(k1,g5,c7,a9,k9)" | ||
| + | contents="R k1 k3 k5 k7 k9"/> | ||
| + | |||
[[category: Open problems]] | [[category: Open problems]] | ||
Revision as of 14:12, 4 October 2021
- 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?
- Seventh row template problem: Does there exist an edge template which guarantees a secure connection for a piece on the seventh row?
- 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 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 are winning, even passing the turn. (This problem was posed by Jory in the 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, and edge template VI1a is such a template.
- 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. Instead, it requires this much space:
The corresponding template of height 9 requires this much space:
