Chains ending in the middle of the board Dots and Boxes

3 replies. Last post: 2008-05-13

Chains ending in the middle of the board
• Dvd Avins at 2008-05-13

For parity, does it matter if a chain ends on the edge or in the middle? Suppose there are the chains, each with one end at the edge and the other at a square with one side filled. When the first chain completes, the second edge of the middle square will be filled, linking the two remaining chains into one. For parity purposes, does the original situation count as 2 chains or 1?

• wccanard at 2008-05-13

I don't quite understand what you mean? Can you make a rubbish ASCII diagram?

But I'll tell you how to answer your own question: the key number is the number of “doublecrosses” made—that is, the number of moves you play that make two boxes with one line. It's the parity of these that people are battling over. You can now work out the rest yourself, as it were. For example a loop counts as 0 chains, because when you keep control with a loop the defender takes 4 boxes, with two doublecrosses. Another example: if there is a square on the board with three long chains running out of it then you might wonder whether this is 2 or 3 chains, but if you think about it there will be 2 doublecrosses so it's 2 chains. If in doubt, count doublecrosses.

wcc

PS you mean “there are *three* chains”, don't you? So in fact the answer I just gave answers your question. For parity purposes the original situation is 2 chains because there will be two doublecrosses.

• Dvd Avins at 2008-05-13

Yes, that's what I meant. Thank you.