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Naccel 2 squad reportback - exercise (a)

Posted By: Matt Ryder
Date: Friday, 15 January 2010, at 6:42 a.m.

In Response To: Naccel 2 -- post #3 (Nack Ballard)

The other day, I made the fatal blunder of skipping ahead of the Naccel tutorials with a question about squads, so Nack responded (rather diabolically) with a little homework to keep me busy:

(a) See how many shift variants (six-syms) you can find for the [333333] stack;
(b) Identify for each which basic squads can be combined to achieve the same formation;
(c) For all the squads, list ways that shifting to the Supers (that flank a five-point field) can reduce the size of the squad or make it disappear;
(d) Count all formations forwards to S0 and backwards to S1; list the partner notations and counts for each.
(e) If you know the (global) count for a squad in one quadrant field, how can you most easily determine its count in another quadrant?

Report back your discoveries for (a), (b), (c), (d) and (e).

Nack opined: "That should quench your thirst for a while :)"

Never were truer words spoken. Since this is an epic undertaking, I've decided to tackle the exercises one by one and split my squad discovery report-backs by question.

This post deals with exercise (a) only.

(a) See how many shift variants (six-syms) you can find for the [333333] stack;

(a) The theoretical maximum width of a six-sym shift for a [333333] stack is 10 pips mirrored around the central point, for a total width of 21 pips. The scope for variants within this range is colossal, so for this exercise, I assumed a maximum symmetry bound of four pips per side around the central point.

In doing this I think I may've made a conceptual error. In his post it seems Nack limits his six-sym shifts here to same-quadrant situations, which I realize (after my investigation, alas) is the implication of this statement:

The number of stack variants (in this case also called six-syms) depends on where the stack is in the field. If it's [111111], there is no variant.

If that's the case, please simply ignore my examples involving points -1, 0 and 7.

Anyway, in this first exploration of six-syms, I deliberately avoided "cheating" by scanning for Nack's squads. Rather I attempted to find as many as possible on my own by simply moving the checkers around the board in as methodical a fashion as I could manage. I probably missed a few, but I think it's a game attempt.

Note that I flailed around trying to describe the structures. Reviewing Nack's posts, I see I've mislabeled some common squads in my descriptions. So please take the words with a pinch of salt and focus on the bracketed items.

Enough caveats. Let's take a look at what I came up with:

Two checker sym-shifts

[233334] - 1 checker either side of a central column of 4 checkers
[133335] - 1 checker either side of a central column of 4 checkers, with a 1 pip mirrored gap
[033336] - 1 checker either side of a central column of 4 checkers, with a 2 pip mirrored gap
[-133337] - 1 checker either side of a central column of 4 checkers, with a 3 pip mirrored gap

Four checker sym-shifts

[223344] - 2 checkers either side of a central column of 2 checkers (six-block)
[113355] - 2 checkers either side of a central column of 2 checkers, with a 1 pip mirrored gap
[003366] - 2 checkers either side of a central column of 2 checkers, with a 2 pip mirrored gap
[-1-13377] - 2 checkers either side of a central column of 2 checkers, with a 3 pip mirrored gap

[123345] - 2 checkers either side of of a central column of 2 checkers - mirrored spread
[023346] - 2 checkers either side of of a central column of 2 checkers - broken 1 pip gap mirrored spread
[-123347] - 2 checkers either side of of a central column of 2 checkers - broken 2 pip gap mirrored spread

[013356] - 2 checkers either side of of a central column of 2 checkers - 1 pip gap mirrored spread
[-113357] - 2 checkers either side of of a central column of 2 checkers - 1 pip gap mirrored scatter
[-103367] - 2 checkers either side of of a central column of 2 checkers - 2 pip gap mirrored scatter

Six checker sym-shifts

[222444] - 3 checkers either side of a 1 pip gap
[111555] - 3 checkers either side of a 3 pip gap
[000666] - 3 checkers either side of a 5 pip gap
[-1-1-1777] - 3 checkers either side of a 7 pip gap

[122445] - Bookend mirrors around a 1 pip gap
[112455] - Reverse bookend mirrors around a 1 pip gap
[012456] - Spread around a 1 pip gap
[-112457] - Broken spread around a 1 pip gap

[022446] - 1pip broken bookend around a 1 pip gap
[002466] - 1pip broken reverse bookend around a 1 pip gap
[-122447] - 2 pip broken bookend around a 1 pip gap
[-1-12477] - 2 pip broken reverse bookend around a 1 pip gap

[-102467] - Broken scatter around a 1 pip gap
[-112457] - Broken spread scatter around a 1 pip gap

[011556] - Bookends around a 3 pip gap
[001566] - Reverse bookends around a 3 pip gap
[-111557] - 1 pip broken bookends around a 3 pip gap
[-1-11577] - 1 pip reverse broken bookends around a 3 pip gap

[-100667] - Bookends around a 5 pip gap
[-1-10677] - Reverse bookends around a 5 pip gap

Matt Ryder.

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