Reasoning through a mate in 2 problem - Part 2

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.

We will be using elimination to cut down the number of White's first moves. Following is the argument structure that I will use to solve this problem:

1. The first move is a queen move
2. The white queen should move such that it controls a1

This will show that the only move that could lead to mate in 2 is 1. Qa8! since that's the only queen move that controls a1.

1. The first move is a queen move

We simply show that the knight on f4 or the king on e1 moving first does not lead to forced mate next move.

If the knight moves, it gives up control of d3. There are two possibilities: the knight moves to e2 or d3 to put the black king in check or it moves somewhere else. For the latter case, it is Black who checkmates White with Nd3#. An example variation is 1. Nd5 Nd3#. The knight move is checkmate because the knight cannot be captured on d3 and the White king has no moves as shown in the figure below:

1. Nd3 is similarly met by 1... Nxd3#. Finally, 1. Ne2+ forces 1... Nxe2. To show that there is no mate now, let's consider all the checks. 2. Kxe2+ or 2. Kf2+ allows the blocking 2... Nd1. On the other hand, 2. Qh6+ allows 2... Nf4. These are the only checks since the queen is the only piece left that can check the king. It can do so only along the diagonal c1-h6 or the rank c1-h1.

To eliminate king moves, we consider the only king move possible in this position, 1. Kf2+. Then, if Black responds 1... Kd2, there is no mate. Black's rook on b1 covers queen checks along the 1st rank. White's knight cannot check the black king in one move because the knight and the king are three squares apart on a diagonal. And if the White queen checks from d5, Black simply blocks it with his knight.

Hence, we only need to consider queen moves for White's first move.

2. The white queen should move such that it controls a1

It's easy to notice that Black is almost in zugzwang. Assuming that White's queen moves somewhere, let's look at Blacks responses abstractly. The black king is immobile — It's blocked by its own pieces and prevented from moving to d1 and d2 by the white king. If the knight on b2 moves anywhere, including to d3, then it does not control d3 anymore. This means that the white knight landing on d3 will result in mate. If the knight on g3 moves anywhere, it cannot control e2 and thus the white knight landing on e2 will result in mate. For this last case, the knight on b2 prevents Black king's escape. The only move left for Black is 1... Ra1.

The impact of knights' moves are illustrated below:

Where are we so far? We have concluded that the white queen should move first and the only "interesting" black response to that is 1... Ra1. Let's look at the following position:

I now show that the White queen is the piece that needs to move again to give mate (after 1... Ra1). Moving the knight on f4 cannot result in mate — First, no matter where the queen is after moving from h1, it still doesn't control b1 (partially because the knight on b2 and the pawn on c2 block important lines). And second, the knight move cannot result in White's control of b1. This is because the knight on a dark square can only move to a light square and from there, it cannot control another light square, such as b1. To complete the argument, the knight move cannot result is the white queen controlling b1 either because the knight move can only clear lines that pass through f4, and none of those lines intersect b1.

Why can't the white king move, resulting in mate? Its only move is to f2. For that to even be check, the white queen needs to be on g1 or f1. None of those control the square d2. To see this, consider the variation, 1. Qg1 Ra1 2. Kf2+ Kd2.

So we have shown that White's queen needs to deliver mate after 1... Ra1. The queen must control both c1 and b1. Given the arrangement of black's pieces, in particular, the knight on b2 and the pawn on c2 that are both defended by the king, the white queen can only control c1 and b1 from a1 itself. Thus, we need to realize the following position after the queen moves to a1 capturing the rook:

This is indeed mate! The white queen is safe, none of the remaining black pieces can block the check, and the king cannot move to a safe square.

So finally, we ask: How can the white queen move from its current position on h1 such that it can control a1 in 1 move? The answer is simply obtained by drawing lines (horizontal, vertical, and diagonal) from both a1 and h1 and seeing where they intersect. The only viable square we get is a8 (for example, h8 doesn't count because the knight on b2 intercepts the diagonal emanating from a1).

Thus, the right first move is 1. Qa8!. 

Conclusion

Here are all the checkmates after 1. Qa8: