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* [https://hexanna1.github.io/hex-study/openings.html Opening explorer] with opening lines on sizes 11&ndash;14 and 17

* [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.

* 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.

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==new templates==

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Fourth row template (IV1e), width 14, carrier size 53 [https://hexworld.org/board/#14c1,f1:pf2:pf3:pf4:pf5:pf6:pf7:pf8:pf9:pf10:pf11a13:pb12:pc11:pd11:pe10:pf14:pg9:ph9:pi9:pj9:pk9:pl9:pm9:pm10:pn10:pn11:p Hexworld link]

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<hexboard size="5x14"

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    coords="none"

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    edges="bottom"

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    visible="area(g1,e2,c3,a5,n5,n3,m2,l1)"

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    contents="R f2 S f5"

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/>

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Fourth row template (IV1f, closely related to IV1e), width 15, carrier size 58 [https://hexworld.org/board/#15c1,f1:pf2:pf3:pf4:pf5:pf6:pf7:pf8:pf9:pf10:pf11:pf12a14:pb13:pc12:pd12:pe11:pg10:pg15:ph10:pi10:pj10:pk10:pl10:pm10:pn10:pn11:po11:po12:p Hexworld link]

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Validity and minimality checked with Mohex.

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<hexboard size="5x15"

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    coords="hide"

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    edges="bottom"

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    visible="area(g1,e2,c3,a5,o5,o3,n2,m1)"

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    contents="R f2 S g5"

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/>

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Validity and minimality checked with Mohex.

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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]

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<hexboard size="6x16"

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    coords="hide"

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    edges="bottom"

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    visible="area(h1,e3,c4,a6,p6,p4,n2,j1)"

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    contents="R g2 S c6"

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/>

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Validity and minimality checked with Mohex.

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I believe this can be extended to an infinite family by adding n adjacent copies of the height-4 corridor marked (*), and [[Edge template V1b]] can be extended the same way; see below. I verified validity and minimality of n=1 for both templates. I have not formally proven the statement for all n, but a proof would likely use the fact that 2-4 parallel ladders can be pushed arbitrarily far before [[Tom's move]].

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<hexboard size="6x17"

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    coords="hide"

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    edges="bottom"

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    visible="area(h1,e3,c4,a6,q6,q4,o2,j1)-l2"

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    contents="R g2 S c6 E *:l3 *:l4 *:l5 *:l6"

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/>

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<hexboard size="6x15"

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    coords="hide"

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    edges="bottom"

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    visible="area(f1,a6,o6,o4,m2,h1)-j2"

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    contents="R e2 E *:j3 *:j4 *:j5 *:j6"

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/>

==basic 1-2* hex strategy==

==basic 1-2* hex strategy==

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