"McGuire the Gathering" Game Sequence

[Dr. Gordon Hamilton]

I'm creating mathematical problems for Canadian schools (see 
http://www.galileo.org/math/puzzles.html). One resulting sequence may be of 
interest to seqfan members:

      McGuire the Gathering

--------------------------------------------------------------------------

      You are the leader of a Scottish clan; either the McGuires or the 
McConnells.

      In preparation for war, you send out your envoys to demand clan 
members come to your aid. Both you and your opponent should secretly write 
down how many days your envoys will travel. If you send them far away, they 
will take longer to return, but will return with more clansmen. 
(Specifically, if you send them X days away, they will return with X men on 
each day starting on the Xth day).

      The winner is the first clan to have 30 clansmen more than their 
opponent.



      For example, if McGuire sends her envoys on a 2 day journey, and 
McConnell sends his envoys on a 7 day journey, then the conflict follows the 
following course:

            Day McGuire
            Clansmen McConnell
            Clansmen
            1
            2 2
            3 4
            4 6
            5 8
            6 10
            7 12 7
            8 14 14
            9 16 21
            10 18 28
            11 20 35
            12 22 42
            13 24 49
            14 26 56

      The McConnell clan wins on the 14th day because there are at least 30 
more of them.

      What is a good strategy?

      Seqfan readers will find that randomly choosing from the group 
{6,8,10,11,12} will produce the best strategy.

      Here are the first 17 such groups if you replace the winning condition 
from 30 in the above problem. Interesting?

            Winning Condition  Group
            1 1
            2 2
            3 3
            4 2, 3, 4
            5 3, 4, 5
            6 3, 4, 5
            7 4, 5, 6
            8 4, 5, 6
            9 3, 5, 6
            10 5, 6, 7
            11 4, 6, 7
            12 4, 6, 7
            13 5, 7, 8
            14 5, 7, 8
            15 5, 6, 7, 8, 9
            16 4, 6, 7, 8, 9
            17 6, 7, 8, 9, 10

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