[Retros] FW: FW: Distance-PG

Francois Labelle flab at wismuth.com
Thu Sep 1 22:20:26 EDT 2016

```Noam Elkies wrote:
> What about *minimum* geometric length?  Here it may
> be possible to prove that one has the absolute minimum (if it's small
> enough that anything beyond the search bounds must be longer).

You're right, it's possible.

The minimum geometric length for a sound checkmate PG is
9.656854 = (4,4,0), so the proof of optimality only required searching
up to ply 9.

Only one sound checkmate PG achieves it:
1. e3 d6 2. Ke2 Qd7 3. Kf3 Qg4#

The second-shortest is
10.242641 = (6,3,0)
1. d3 g6 2. Qd2 g5 3. Qxg5 f6 4. Qh5#
(also the only solution).

So I guess someone could ask:

"Compose a sound checkmate PG with geometric length less than 10".

The problem has the following properties:
- the solution is a chess game,
- the solution is unique,
- the statement doesn't talk about the number of moves. In particular,
it doesn't specify it (like a PG stipulation does), nor does it ask to
minimize it (like an SPG stipulation does).

Is this the first known example of a chess problem with these properties?

For completeness, the minimum geometric length for a checkmate *game* is
8.414214 = (7,1,0)
with two solutions that I've already mentioned:
1. d3 e6 2. Qd2 e5 3. Qe3 Ke7 4. Qxe5#
1. d3 e6 2. Qd2 Ke7 3. Qe3 e5 4. Qxe5#

François
```