# [Retros] FW: FW: Distance-PG

Noam Elkies elkies at math.harvard.edu
Thu Sep 1 23:14:06 EDT 2016

```I 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).

Francois Labelle <flab at wismuth.com> replied:

> 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: [...]

Great!

andrew buchanan <andrew at anselan.com> writes:

> > The second-shortest is 10.242641 = (6,3,0)

> > So I guess someone could ask:
> > "Compose a sound checkmate PG with geometric length less than 10".

> This unnecessarily removes the case 10.

> I would prefer the more whimsical stipulation:"Compose a sound
> checkmate PG with geometric length less than 10.24264."

Normally "geometric length at most 10", though "length less than
6 + sqrt(18)" hints at the runner-up length too.

NDE
```