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Re: X/O Grid
Posted:
Dec 23, 2012 1:17 PM


On Sun, 23 Dec 2012 00:16:38 0800, William Elliot wrote: > On Fri, 21 Dec 2012, James Waldby wrote: >> On Tue, 18 Dec 2012 22:32:43 0800, William Elliot wrote: >> >> > Consider an n by m grid of x's and o's. > > n is width, m is height. > >> > An xpath from the bottom squares of the grid to the top squares of >> > the grid constitutes a sequence of horizontally or vertically adjacent >> > xsquares from some bottom xsquare to some top xsquare. Similar >> > with opaths. >> > >> > An exit is a top square that is connected to a bottom square by >> > either an xpath or an opath. For example, in the 8 by 3, x/o grid >> > >> > x x o o x x o o x >> > x o o x o x x x x >> > o o x x x o x o x >> > >> > has two oexits and three xexits for a total of five exits. >> > >> > In an n by m, x/o grid, what is the expected number of exits? >> ... >> <http://pat7.com/jw/pathcounts/sampleresults> for more lines). >> >> m,n: 3 2 Total: 54 Avg.: 0.84375 >> m,n: 3 3 Total: 1194 Avg.: 2.33203125 >> m,n: 3 4 Total: 18306 Avg.: 4.46923828125 >> m,n: 3 5 Total: 231634 Avg.: 7.06890869141 >> m,n: 3 6 Total: 2614194 Avg.: 9.97235870361 >> > Yes, as the width n increases, the expected number of exits will incease. >> ... >> The other program, expCount.py, computes data like the following >> via a few milliseconds of work. >> >> p= 0.5 >> m n= 2 3 4 5 6 7 8 >> 3 0.8438 2.3320 4.4692 7.0689 9.9724 13.0664 16.2766 >> 4 0.6328 2.1489 4.6596 7.9910 11.9236 16.2681 20.8833 >> 5 0.4746 1.9803 4.8587 9.0363 14.2647 20.2720 26.8264 >> 6 0.3560 1.8249 5.0666 10.2193 17.0690 25.2702 34.4792 >> 7 0.2670 1.6817 5.2834 11.5575 20.4259 31.5051 44.3256 >> 8 0.2002 1.5498 5.5094 13.0711 24.4436 39.2805 56.9899
> I don't see why as the height m increases, that the expected number of > exits would increase. In fact, I'd expect just the opposite.
Yes. What the programs are counting are the number of expected paths rather than the number of expected exits. I imagine I can revise the programs to count exits rather than paths.
Percolation theory, mentioned by Michael Press, may provide some answers for exits or paths. <http://en.wikipedia.org/wiki/Percolation_theory>
>> Note that the paths considered by the two programs only continue >> (from a given square) by going left, right, or upward to a >> notyettraversed square. That is, they count the following grid >> as having no paths of either kind: >> >> 0 0 0 0 1 >> 1 1 1 0 1 >> 1 0 1 1 1 >> 1 0 0 0 0 > > Oh oh. That grid has one exit. Note that the grid > > 1 1 1 0 1 > 1 0 1 1 1 > 1 0 0 0 0 > > has four exits. [rather than only three]
Allowing downmoves would complicate the programs quite a bit, so I left them out. Also, your original post didn't have examples or text about downmoves (vs left, right, or upmoves) from a square so it wasn't clear if they were to be allowed. I'll have a look.
 jiw



