Test your chess math skills!

Today is the first day of the new Hebrew Year, 5778. A mathematically auspicious number!

In an entirely unrelated matter here is a chess position.

Assuming optimal play, how many different possible sequences are there of the next 18.0 moves?

(Shorter games don't count. Ignore the rules for drawing by repetition & dead position.)

Good luck solving!
Andrew

This is rather difficult. I am lazy to compute the number out. There are many things to consider, including en passants and the stuck pieces being able to move at a later time.

The easier path of 1. Kf1 bxc2 is easier to compute because Black has no choice but to oscillate between two squares while the White king is confined to four squares. So in this case, essentially this problem reduces to a problem of how many ways the White king can move.

So what happens after 1. Kf1 dxc2 2. Ke1 when the only Black piece which can be moved, the king, gets captured after any move?

So, there are two first moves: 1. Kd1 or 1. Kf1. In either case, Black can then play either 1... bxc2 or 1... dxc2. Let's figure out what happens.

First, 1... bxc2 completely blocks the game. The game is drawn here (a dead position; no player can possibly checkmate the other), but since we ignore the dead position rule, we'll just shuffle kings back and forth until we reach 18.0 moves. Can Black do better than drawing by taking 1... dxc2?

If the first two plies are 1. Kd1 dxc2+, then White needs to escape check with 2. Ke1 or 2. Ke2; in either case, Black is stalemated, so this is a sequence of 1.5 moves, not 18.0 moves. So Black can take a draw as well, but it doesn't give additional sequences.

If the first two plies are 1. Kf1 dxc2, what happens?

...well, probably not. All Black's moves are forced (except the 1... dxc2 at the beginning), and now that White king can get out, one White pawn should promote. So I'm pretty sure Black loses here, and so Black won't take 1... dxc2.

So Black takes 1... bxc2. Now, Black can only shuffle the king on a4-b3, and White can only shuffle the king on e1-f1-g1-h1. Black clearly doesn't have any move left, but what about White?

After the first move, White king is either on d1 or f1. After the second move, there are 2 sequences where White king is on e1 (coming from d1 or f1), and 1 sequence where White king is on g1 (coming from f1). Then we can continue further:

• Third move: 3 sequences on f1, 1 sequence on h1
• Fourth move: 3 sequences on e1, 4 sequences on g1
• Fifth move: 7 sequences on f1, 4 sequences on h1
• Sixth move: 7 sequences on e1, 11 sequences on g1
• ...
• 18th move: 2207 sequences on e1, 3571 sequences on g1

I'll let you verify it, as well as figuring out any better way to compute it. In either case, the total number of 18.0-move sequences is 2207+3571 = 5778, the exact value of the current Hebrew Year.

Well done Mr Chaotic. Just like the Fibonacci sequence 1,1,2,3,5,8... there is a Lucas sequence 1,3,4,7,11... In both, each term is the sum of the two before. You can set up a Fibonacci sum on a chess board by having K or B wander around a cell of 4 squares in a line. This problem is to show that you can do get the Lucas sequence if you close the door on the king as he's trying to get in. 5778 is a Lucas number. So much, so quite clean, the issue is that the stipulation and the left hand side of the board are real messes. Working on how to tidy that up...