01.25.16 In a previous article, we presented you with a confounding problem of the proof game genre and asked you to compose a game that ends with the move 6...Nf1 mate. The time has come to reveal the solution to this Gordian knot. We will discuss the number of possible paths to a solution and share some comments we received from our faithful readers. Warning! If you would still like to have a go solving this problem, do not read this post!

There are 16 possible final solution positions to the 6...Nf1 mate problem. We will justify this number later on in the article as it is easier to visualize and understand it after having a look at one of the possible solution positions.

First, a bit of chess mathematics. In the initial position, each side has a choice between twenty options. Each pawn can move one or two squares and each Knight has two possible moves. We can write this mathematically in the following manner: [8 (the number of pawns) x 2 (the number of moves each pawn has)] + [2 (the number of Knights) x 2 (the number of moves each Knight has)] = 20. In chess, we call each half-move a ply. So after each side makes a move, two ply have elapsed. Following the logic of the above math, after each side makes just one move there are 400 (20 x 20) possible positions that can occur. So far so good. Continuing in this way we can say that after three ply there are 8,902 possible positions and after four ply (two moves from each side) there are 197,742 possible positions. As you can see, the number of possible positions after a small number of moves for each side becomes astronomical in short order.

When faced with this conundrum, American mathematician Claude Shannon, in an attempt to come up with a figure for the number of possible positions in a chess game after 40 moves for each side (80 ply), posited that, on average, the number of possible moves available to each player in a given position is 30. In some positions there are more possible moves and in some less but 30, Shannon argued, is a valid estimate. For example, after 1.e4 e5 2.Nf3 Nc6 White has 27 moves available. And after 1.e4 e5 2.Nf3 Nc6 3.Bb5 Nf6 White has 32 possible moves available. Using 30 as an approximate number of move choices available to each player on each turn we can continue the exercise of calculating how many possible positions that can arise after each ply and each move set (two ply).

Recall that after four ply the number of possible positions was 197,742. Using Shannon's estimate of 30 possible move choices on each turn, it follows that after five ply the number of possible positions would equal approximately 197,472 x 30 or 5,932,260. After six ply (just three moves per side), the number of possible positions swells to 177,967,800. That's 5,932,260 x 30.

Thus:

after seven ply there are approximately 5,339,034,00 possible positions;
after eight ply (four moves for each side) there are approximately 160,171,020,000 possible positions;
after nine ply there are approximately 4,805,130,600,00 possible positions;
after ten ply (five moves for each side) there are approximately 144,153,918,000,000 possible positions;
after eleven ply there are approximately 4,324,617,540,000,000 possible positions;
and after twelve ply (six moves for each side) there are approximately 129,738,526,000,000,000 possible positions.

That's one hundred twenty-nine quadrillion, seven hundred thirty-eight trillion, five hundred twenty-six billion possible positions after six moves for each side.

Hopefully, this will give your conscience some solace if you struggled with this chess composition problem.

Shannon wrote the following formula to calculate the approximate number of positions that can arise in a 40-move (80-ply) game:

N = 10*e3*e40 = 10*e120

The Shannon Number, 10*e120 or 10 raised to the 120th power, is a rough estimate of the number of positions that can arise in a 40-move game. As a comparison, it is estimated that there are 10*e80 or 10 raised to the 80th power atoms in the observable universe. You could therefore assign billions of games of chess to each atom in the observable universe and still not have exhausted all of the possible positions in a 40-move game. By the way, the longest tournament chess game (in terms of moves) ever to be played was Nikolić-Arsović, Belgrade 1989, which lasted for 269 moves.

Here is the paper in which Shannon first introduced the figure later to be known as the Shannon Number Programming a Computer to Play Chess.

But we digress.

Here is one of the 16 possible solution positions. This position arose after the moves 1.e4 e5 2.Bc4 Nf6 3.Ne2 Ng4 4.0-0 Qh4 5.Kh1 Nxh2 6.Rg1 Nf1 mate.

White is mated.

After some consideration, we can see that the possible choices we have to vary the diagram and still have a position that ends with 6...Nf1 mate, involve the two pawns on the e-file and the White Bishop. All of the other pieces need to be in precisely the locations that we see in the diagram.

Therefore we arrive at the 16 possible solutions to this problem using the following reasoning and calculations. All Black's pieces, except the e-pawn, must be on the squares shown in the diagram. Black's e-pawn has the choice of two squares: e6 or e5. With Black's e-pawn on e6, White's e-pawn can be on e3 or e4 and White's Bishop can be on d3, c4, b5, or a6.

We will calculate the number of possible positions that result in 6...Nf1 mate with Black's e-pawn on e6 first, then do the same with Black's e-pawn on e5 and, finally, sum the results to get the total number of acceptable positions that serve as solutions to this problem.

Here goes.

Black's e-pawn on e6.

White's e-pawn on e3 and White's Bishop on either d3, c4, b5, or a6. Four possible positions
White's e-pawn on e4 and White's Bishop on either d3, c4, b5, or a6. Four possible positions

Black's e-pawn on e5.

White's e-pawn on e3 and White's Bishop on either d3, c4, b5, or a6. Four possible positions
White's e-pawn on e4 and White's Bishop on either d3, c4, b5, or a6. Four possible positions

Summing all the possible positions with Black's e-pawn on e6 and e5, we conclude that there are 16 possible positions that can occur that end with 6...Nf1 mate.

Here are some comments we received from our readers.

I could do it in 7 without ending on a capture, but I couldn't find a way to do it in 6. Had to look it up.

1.e4 Nf6 2.Qh5 Nxh5 3.Ba6 b6 4.Ke2 Bxa6+ 5.Ke3 Ng3 6.Nf3 e5 7.Rf1 Nxf1+
This ends with 7...Nxf1 mate and is therefore incorrect.

1.d4 e5 2.f3 Ee7 3.Nd2 Kf6 4.g4 Kg5 5.Bg2 Kf4 6.Nf1 mate
A fantastic try that ends with 6.Nf1 mate (not 6...Nf1 mate)

I could only do it in 7 moves (7...Nf1#); so I am curious about a solution to this formidable brainteaser.

AHAHAHAHA! Finally got it!
A correct solution was then given.

Finally, now I can sleep. I had tried everything but castling and I fell into this line. No brilliancy here believe me just grunt work.
A correct solution was then given.

I'm mad at you!

I had to Google it!