Pandolfini's Puzzler #24 - The 50-50 Problem

Lucian: I guess a hard decision would be where one has many choices.

Zephyr: I’ve heard you say that when you have many choices it can be like having no choice at all. You might get lost in all the possibilities. 

Professor: So does that mean the easiest type of decision to make is when you have but one possibility?

Lucian: “Yes,” I think so. But even with a single choice I might not like the consequences.

Zephyr: Yeah, the consequences could be very bad. You might be in zugzwang or something.

Professor: That’s funny. I’ll have to add that to my repertory of jokes. 

Lucian: Professor, I didn’t know you like to tell jokes.

Everybody seemed to smile.

Professor: I do, before I tell another joke, I’d like to clarify something. In some cases, if you have a single choice, or a single move, you don't really have any decisions to make. I tell you what. Let’s look at a position with Black to play. Here, there aren’t many moves (or choices). Nor is it a position with just one move. Rather, it’s a position where Black has two moves.

Professor: Again, it’s Black’s turn, and clearly he has two possibilities. He can move his king to h1, shifting it apparently farther away from the action. Or he can move his king to f1, keeping it apparently closer to the action. There’s a wrong way to go. And there’s a right way to go. The wrong way loses. The right way draws.

Zephyr: Wow! Black has a 50 percent chance of drawing.

Lucian: Which means, he has a 50 percent chance of losing. 

Professor: That’s why I’m going to call this the 50-50 problem.

Question: How can Black move and draw? 

Professor: Or, restating the question, should the black king move closer to the action or farther away?

Zephyr: From the way you’ve set that problem up, Professor, I feel you’re being a little tricky.

Lucian: Maybe it has something to do with what you mean by “closer” and “farther.”

Professor: Perhaps you’re right, Lucian. Anyhow, can either of you, or the both of you, tell me which move is right, and which move is actually closer? By the way, you’re going to have to explain why.

The right move is 1…Kh1! But before we see why, let’s examine what happens if the king instead moves to f1. 

After 1…Kf1, let’s say White plays 2. Kf3. 

If Black tries 2…Ke1, play might continue 3. Ke4 Ke2 4. Kd5 Kd3 5. Kc6 Kc4 6. Kb7 Kb5 7. Kxa7 Kc6 8. Kb8, and the pawn will queen.

Or if Black tries 2…Kg1, play might continue 3. Ke4 Kf2 4. Kd5 Ke3 5. Kc6 Kd4 6. Kb7 Kc5 7. Kxa7, and the pawn soon queens anyway.

Now let’s look at what happens after the right move, 1…Kh1!

After 1…Kh1, a possible line is 2. Kf3 Kh2 3. Ke4 Kg3 4. Kd5 Kf4 5. Kc6 Ke5 6. Kb7 Kd6 7. Kxa7 Kc7, and the position is drawn, since 8. Ka8 Kb6  9. a7 Kc7 is stalemate!

It’s not always obvious which path is closer or farther away on a chessboard. By trying 1…Kf1, it looked on the surface as if Black were moving the king closer to the action. But it was in fact moving the king farther away from something else: the key diagonal of retreat, that is, the b8-h2 diagonal. That diagonal contains the c7-square. Immediately after the a-pawn falls, Black’s king can set up a draw by occupying c7. White’s king is then unable to escape from the corner. In other words, by moving farther away, Black was actually moving closer.

RELATED STUDY MATERIAL

• Here's another draw in king and pawn endgames that you should know;
• Or, if you prefer, a more detailed explanation in video form;
• If you want a real challenge, do Benzoo's king and pawn endgame course - parts one, two and three!