Dots and boxes game in CGT

Dots and boxes game in cgt

Here’s a compact CGT (combinatorial game theory) view of Dots & Boxes—enough to analyze real endgames and pick the right move.

1) Model: split into independent components

In the typical endgame every remaining region is either a:

• Chain: a path of (k) boxes with two open ends (length (k\ge 3)).
• Loop: a cycle of (k) boxes (even (k\ge 4)).

The total game is (approximately) a disjunctive sum of these components. Each component is a “loony” game: if you take it, you also keep the move, which is why control matters.

2) Controller vs. Opener and the hard-hearted handout

• The player who is forced to open a component is the opener.

• The other player tries to keep control by doing a hard-hearted handout (HHH):

  • In a chain (k\ge 3): take (k-2) boxes, then hand 2 to the opponent, keeping the move.
  • In a loop (k\ge 4): take (k-4), then hand 4, keeping the move.

This repeats until the very last component, where the controller may simply finish everything.

3) Controlled value (quick evaluator)

Let your position (P) have chains (c_i) and loops (l_j). Define the controlled value [ \mathrm{CV}(P)=\sum_i(c_i-2);+;\sum_j(l_j-4). ] Intuition: this is how many more boxes the controller nets (vs. the opener) if they keep control all the way with HHH.

Rule of thumb (very strong and easy):

• If (\mathbf{CV(P)\ge 2}), optimal play is to keep control throughout; the margin equals (\mathrm{CV}(P)).
• If (\mathbf{CV(P)\le 1}), exceptions arise: sometimes it’s best to give up control deliberately (not use HHH) to swing parity—most famously with 3-chains.

4) Move-choice (what to open?)

When you’re the opener (you must start a component):

1. Temperature heuristic: Loops are “hotter” than chains (roughly: loops (\approx) temp 2, chains (\approx) temp 1). If you must open something, open a smallest loop first; if no loops, open a smallest chain.

2. Low-CV exception play ((\mathrm{CV}\le 1)):

   • If there’s a 3-chain, often take all 3 (don’t hand out 2). That flips control and can improve your final margin.
   • If multiple 3-chains exist, parity of the count matters: with an odd number of 3-chains you can frequently engineer a +1 swing by timing when you give up/keep control.

3. Otherwise ((\mathrm{CV}\ge 2)): open the smallest component of the hottest type and expect the controller to HHH; your job is mostly damage control.

5) Worked micro-example

Position: chains (3,3,4) and one loop (6).

• Sum of boxes: (3+3+4+6=16).
• (\mathrm{CV}=(3-2)+(3-2)+(4-2)+(6-4)=1+1+2+2=6.) Since (6\ge 2), controller keeps control with HHH everywhere and wins by 6 boxes. (You can also see it as “giveaways”: (2+2+2+4=10) boxes handed out; controller gets (16-10=6).)

6) Practical endgame cheat-sheet

• Count CV quickly. If (\ge 2): keep control (as controller) or open smallest/hottest (as opener).
• 3-chains are special when (\mathrm{CV}\le 1): the opener often takes all 3 to break control.
• Loops beat chains for opening order (they’re hotter).
• Avoid creating new 3-chains earlier in the game unless you know the parity helps you.

7) Further study (classic sources)

• Elwyn Berlekamp’s analysis of “strings and coins” (equivalent to Dots & Boxes endgames).
• “Winning Ways for your Mathematical Plays” (Conway, Berlekamp, Guy) for CGT temperature and thermographs.
• “The Dots-and-Boxes Game: Sophisticated Child’s Play” (Berlekamp) for a complete treatment of chains/loops and parity tricks.

If you want, give me any specific endgame (list its chain and loop lengths), and I’ll compute the margin and tell you the exact best first move.