Hi Colin,
coloin wrote:blue wrote:I did some testing with random grids.
10,000,000 samples produced 461 grids with a 20 and no 19.....
....
Is this an upper bound ? - or have you actually checked [with a "checking program"] that those 461 grids didnt have a 19 ? [i get the feeling you have]
Yes, and for good measure, I checked them to make sure they had a 20, too.
coloin wrote:i'm thinking probability-wise that the number of grids without a 20 is a lot less than 164 ... maybe only 4 ?
It's certainly possible.
For "automorphic" grids the ratio of "no 19" to "no 20" grids was ~1200:1.
I didn't know what to expect, but I wondered if that might mean there were another ~4800 "non-automorphic" grids with no 20.
It's clear now, (for me, anyway), that there are nowhere near that many !
anyhow ive done a bit of research whilst mathemagic's program is ongoing ......
taking 8 random 18s - i generated similar pattern template 18s ... there appears to be a lower bound of an average of 26 puzzles per pattern
taking 8 random 19s - i generated similar pattern template 19s ... there appears to be a lower bound of an average of 10000 puzzles per pattern ... new ones still appearing though at the {+5} level
This makes the ratio between patterns with valid 18C puzzles / possible ED 18C patterns at smaller than 1/1000 which is difficult to understand if its correct !
The ratio is more like 1/95, using
JPF's count of 184,060,159,680 ED patterns, and my estimate of 1,934,000,000 18C puzzles (mentioned earlier in the thread).
It seems like you've shown that if an 18C/19C pattern has one puzzle, then it's many times more likely that the same pattern has another puzzle, than it is that a randomly chosen pattern of the same size, has at least one puzzle.
ive been looking at a grid "which has an 18" but i've been struggling to find the 18 puzzle from scratch ....
this grid
- Code: Select all
347981256582476193169523874896245317754318962213769485925137648478692531631854729
how long does your program take to confirm one 18-puzzle ? [ and how many 19s ? !]
5.4 seconds to confirm "only one" 18, and 213 seconds to find 2709 19's (including the 63 non-minimals).
(Any confirmation(s) of the 2709 number ?)
Cheers,
Blue.