### 2x2 corner puzzle Dots and Boxes

5 replies. Last post: 2007-09-05

2x2 corner puzzle
• wccanard at 2007-08-30
``````6 + + + + +-+      |   |b|5 +-+-+-+-+-+  |b|b|b|b|b|4 +-+-+-+-+-+  |b|b|b|b|r|3 +-+-+-+-+-+      |r|r|r|2 + + +-+-+-+      |r|r|r|1 + + +-+-+-+  a b c d e f
``````

I just “analysed” the 2x2 corner in the same way I analysed the 3x2 corner (see a recent “puzzle” thread). Here's a cute one. Blue is 3 boxes up and it's his move. How does he win?

wcc

Blue moves a5-a6.

I think this puzzle is much easier than the 3x2 puzzles. In fact, saying that Blue has a win is almost a hint!

• ypercube ★ at 2007-09-05

after Blue:a5-a6, Red plays a2-b2. How does Blue win there?

Blue replies with b1-b2.

wc – I think that these puzzles deserve ratings!

• wccanard at 2007-09-05

@sam: Correct on all counts (probably also including the comment about the ratings). I find it quite counterintuitive that to minimise the length of the chain, blue's best idea is to stay well away :-)

Regarding ratings: think about it from my point of view. I did not hand-generate these puzzles. I created 80=5*2*2*2*2 puzzles in the following way: choose one of (2x2,3x2,4x2,3x3 icelandic or 5x2 edge) and then (1-chain or not),(2-chain or not),(icelandic 2x1 or not),(very long chain or not). I solved all 80 puzzles by computer and looked at all 80 answers. And for about 60-70 of them I thought “yup, I would have done that, well, probably” or “wooah, almost every natural move is a win here”, but for a for a few I thought “hmm, I probably wouldn't have done that at all” or “wow, my first impression is that it's hard to believe that this move is the unique winning move” [for example the puzzle above!], and I posted those ones. I don't have any real idea about how hard they would be for a human to solve. Perhaps the most naive comment is the obvious one: as the open regions get bigger, giving rigorous solutions will surely get exponentially harder. So, for example, for the 5x2 regions I don't expect someone to post saying “I've done this by hand, blue's unique winning move is blah and here are his cunning responses to the following 10 replies by red”—this is asking a lot of a human. I certainly wouldn't be able to do any of those 5x2 puzzles on a rigorous level, and I'm not sure any human would, unless they have a lot of time on their hands and a lot of patience! But I will *definitely* accept responses in the bigger puzzles of the form “what about if I try here? What do you do then?“. The bigger regions perhaps aren't really puzzles at all, more like interesting positions about which I am happy to talk with the hindsight of 10 seconds of computer analysis :-) On the other hand that 3x3 with the 11-chain in, I think that I would have had some chance of doing that one—given the hint that it's possible, as you say above!

wcc