Reasoning through a mate in 2 problem - Part I

In this article, I will be discussing a mate in 2 puzzle composed by Sam Loyd and included in the Kindle e-book "100 mate in 2 chess puzzles".

Introduction

The puzzle is given in the diagram below with White to move and checkmate in 2. Feel free to spend some time thinking about it yourself before you read the rest of the article.

I will derive the right first move using logic. The main technique I will be using is elimination: It is simply not possible to examine every first move that White can make, therefore, we strive to get the number of candidate moves down to a manageable number.

What follows next is the proof structure. I will be proving each statement separately.

1.  White's first move must prevent Black from moving his king to e7
2.  White's first move is either 1. Qc5 or 1. Qb4
3.  White's second move if Black is able to play 1... g4 must be 2. Qe4#

From point 2 and 3, it follows that the first move must be 1. Qb4 since the queen cannot move to e4 from c5. Let's prove the statements now.

1. White's first move must prevent Black from moving his king to e7

We will prove this by contradiction. Suppose White makes his first move and then Black plays 1... Ke7. First, note that after 1... Ke7, the only White piece that can deliver checkmate is the queen since the pawn on d3 cannot reach d6 in two moves. Now, White's first move can be either with the king, or the queen or the pawn. If it is with the latter two, the final queen move must control e7 and the surrounding squares of e8, e6, and f8. This is shown in the diagram below.

It's an easy visual exercise to confirm that the only square for the White queen to achieve the above objective is f7, but the Black king controls f7! To solidify this idea, consider the variation, 1. Qc4+ Ke7 2. Qxf7+ Kxf7. Therefore, no checkmate is possible if White moves his queen or the pawn on the first move in a way that allows Black to play 1... Ke7. Another important point to note is that the queen does not control all the four squares from e7 itself because a piece on e7 does not control e7!

Finally, if the White king moves in a way that allows 1... Ke7, the problem remains the same. The White king still cannot control either of e8, f8, or e6, so the queen must do that on the next move, which we have verified is impossible.

2. White's first move is either 1. Qc5 or 1. Qb4

We have shown that White must disallow 1... Ke7. Let's filter down to only the moves that achieve this objective. The moves are 1. Kd8, 1. Qd7+, 1. Qc5, 1. Qb4, 1. Qe5. To find the four queen moves, the simplest is to extend diagonals, horizontal, and vertical lines from e7 and check if the White queen can move there from b5.

Now, let's filter these further. 1. Kd8 allows 1... Kd6. Now, the White queen cannot move in a way that controls c6, d6, e6, c5, d5, and e5. Next, after 1. Qd7+ Ke5, there is no mate. And obviously, 1. Qe5 is simply met by 1... fxe5.

Thus, we are left with only two queen moves, to c5 and b4. At this point, we can simply check the two moves one by one. But, there's another independent observation we can make that makes the decision obvious.

3.  White's second move if Black is able to play 1... g4 must be 2. Qe4#

Neither of 1. Qc5 or 1. Qb4 prevent 1... g4. The second move again in this case must be a queen move. The queen needs to control e6, e7, e5, and f5. It's illustrated in the following diagram:

The first three squares are consecutively placed on the e-file. This leaves very few choices for controlling these: d6, f6, and any other square on the e-file. Combining that with the need to control f5 means that the queen must be on f6 or e4. But, f6 is controlled by the Black king too. Therefore, 2. Qe4# must be the response to Black playing 1... g4.

This shows that 1. Qb4 is the right move since the queen cannot move to e4 from c5.

Conclusion

1. Qb4 is the solution! Here are all the checkmates: