Chess puzzler

Re: Chess puzzler

Jim Rutten

I have often wondered why this wouldn't work but not having played chess since high school there is probably some rule against it. If I was playing for a draw in 2 move chess I would move right knight out and back and then left knight out and back ad nauseum.

On the right track

Alex Y

The move you describe is key to the answer, but as you describe the game, what's to stop Black from moving his men across the board and putting you in check?

Hint

Alex Y

For you math types, the answer comes with a non-constructive existence or non-existence proof.

In English: you can answer the question and show why your answer is right without showing White's strategy for avoiding loss or Black's strategy to assure a win.

And don't forget that Jim's move is important.

Question

Bill Earl

Are both moves with the same piece? Or can a player make two moves using two different pieces per turn?

Re: Question

Alex Y

They are allowed to move two pieces, or a single piece twice.

Re: Question

Bill Earl

After peeking at your hint, I guess my question was irrelevant.

It is not possible for white to not have a non-losing strategy. If we assume no non-losing strategy, then nudge the knight in a self nullifying manner, now the new state is no different than the old which would necessitate that neither black nor white are non-losers which we know is a non-possibility.

Furthermore, we can conclude that moving the knight for naught is not non-losing.

(and they say I have a negative attitude )

  Correct

Alex Y

Although there were so many negatives in that answer that I got tied up and thought you had it backwards .

Except that

Alex Y

moving the knight for naught is not non-losing.

Re: Except that

Bill Earl

Is it? White starts by having a non-losing strategy. By making a null move, white effectively passes and gives black a non-losing strategy. Giving your opponent an opportunity for not losing is "not non-losing" on your own part.

Don't forget ties

Alex Y

After the first move, Black has a non-losing strategy, but not necessarily a winning one. They can both have a non-losing strategy; they just cannot both have a winning strategy.

Correction

Alex Y

I think I overstated the case.

I think all you can say is that White's move (pawn out and back) MIGHT be a non-losing move. We know that White started with a non-losing position, and after that move, leaves Black with a non-losing position, but we do not know whether either has a winning strategy (which is equivalent to the other not having a non-losing strategy).

Or not. My brain is tired!

Re: Correction

Jim Rutten

Wait a minute. You can't move pawns up and back. I thought we still had to go by the rules of chess and pawns can only move forward (at least until the reach the other side to be traded for another piece). Hence, my thought about knights moving out and back (a perfectly legal move). But, there is one spot on the board where the opponenet's knight can be moved to put my king in check and I cannot take it with my knight (if I am playing black and the opponent is playing white it would be E6). Then I would have to take the knight with a pawn, and the realm of possibilities flood in and out the door with my non-losing strategy. Therefore, while staying within the rules of the game (pieces only moving legally) I would submit to you, Alex, that there is no non-losing strategy. Please, please prove me wrong so I can stop thinking about this but you will have to do so a piece movement strategy not double (or triple or quadruple) negatives.

Re: Correction

Alex Y

First of all, that mention of pawn movement was a brain fart on my part . I meant the knight up and back, as you mentioned.

As far as the proof, I'm afraid it can't be done (at least by me!) constructively -- I can't show you what White's strategy is. All I can do is show you that it exists. I'll try rewording Bill's proof with fewer negatives, but I can't totally eliminate them.

At the beginning of the game, there are three possible outcomes --White wins, Black wins, or they tie.

First, let's call a "non-losing strategy" an "at-least-a-tie" strategy -- that gets rid of one of the negatives. If White has an at-least-a-tie strategy, that means that he is able to play in such a way that Black cannot win.

Hypothesis: White does not have an at-least-a-tie strategy.



That would mean that whatever moves White makes, Black will always be able to win. Specifically, if White makes one of the double knight moves returning the board to its original state, then Black would be able to win. But Black is looking at the same board as White originally had (except for the position of the K and Q, which you can convince yourself is irrelevant by playing while looking at the board in a mirror).

So our hypothesis that White does not have an at-least-a-tie strategy leads to the conclusion that Black, looking at the identical setup, has a winning strategy, which is a contradiction.

Because that hypothesis leads to a contradiction, it must not be true. Hence, If it is false that White does not have an at-least-a-tie strategy, that means it is true that White does have such a strategy.

I don't know if that is any clearer, but I'm afraid that is my best shot at it.

Thanks for the explanation ...

Larry Barrett

I was going down the same path as Jim, and getting nowhere. And thanks for the puzzle.

Re: Thanks for the explanation ...

Jim Rutten

I agree with Larry. Thanks for the puzzle. It has been a while since I have had smoke running out of my ears.