This file contains some manually-obtained results, including
manually-designed algorithms for a few cases.  All algorithms herein
have an implicit "if the bell doesn't ring" attached to each step.

Because any algorithm can be expressed in this fashion, if there is no
algorithm of this sort, there is no algorithm at all.  Thus, if the
program says no algorithm was found, no algorithm exists unless it's
buggy.

W prime >3 and 2<=H<=W-2 suspected impossible.  Not "W odd >3 and
2<=H<=W-2"; W=9 H=7 is a counterexample.

W>2 H=W-1 borders on trivial.

W=2 H=1 - trivial.

W>2 H=1 - easily proved impossible.  (It is possible for every spin to
cause the single probe to miss the same two wells, which can be
different.)

Conjecture: W=w H<w/2 is impossible for basically the reason described
for W=6 H=3: every pattern has either -X- or --- in it.

W=4 H=2 - the standard puzzle-book version.
2 patterns: XX-- X-X-
	XX--: Both down
	X-X-: Both down
	X-X-:
		If one up: invert it, ring
		If not: invert either
	XX--: Invert both
	X-X-: Invert both

W=5 H=2 - easily proved impossible; two cells *-*-- can be different
and fall into unprobed positions on every spin.
2 patterns: XX--- X-X--

W=5 H=3 - manually suspected impossible, computer search agrees
2 patterns: XXX-- XX-X-

W=6 H=3 - easily proved impossible; two cells *-*--- can be different
and fall into unprobed positions on every spin, because every pattern
has either -X- or --- in it.
4 patterns: XXX--- XX-X-- XX--X- X-X-X-

W=6 H=4 - possible
3 patterns: XXXX-- XXX-X- XX-XX-
	XXXX--: all down
	XXX-X-: all down
	(Now have *-----)
	XX-XX-: if any u, invert it, ring
		if not, invert **.--.
	(Now have ***---)
	XXXX--: three u, invert them, ring
		three d, invert them, ring
		else, invert *--*..
	XXX-X-: invert *-*.*., ring
this pattern is generalizable to all W even >6, H=W-2; see 8/6 and 10/8
for examples.

W=7 H=2 - impossible, basically same proof as W=6 H=3
3 patterns: XX----- X-X---- X--X---

W=7 H=3 - impossible, basically same proof as W=6 H=3
5 patterns: XXX---- XX-X--- XX--X-- XX---X- X-X-X--

W=7 H=4-5 - manually suspected impossible, confirmed by program

W=8 H=2 - impossible; 8/3 is impossible

W=8 H=3 - imposible, basically same proof as W=6 H=3
7 patterns: XXX----- XX-X---- XX--X--- XX---X-- XX----X- X-X-X--- X-X--X--

W=8 H=4 - possible
10 patterns: XXXX---- XXX-X--- XXX--X-- XXX---X- XX-XX--- XX-X-X-- XX-X--X- XX--XX-- XX--X-X- X-X-X-X-
No algorithm known except for decision tree from program.  This is
bzip2 | btoa on that:
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xbtoa End N 1045 415 E af S 20e35 R 6f455144

W=8 H=5 - possible
7 patterns: XXXXX--- XXXX-X-- XXXX--X- XXX-XX-- XXX-X-X- XXX--XX- XX-XX-X-
No algorithm known except for decision tree from program.  This is
bzip2 | btoa on that:
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W=8 H=6 - possible; generalization of W=6 H=4
	XXXXXX--: all down
	XXXXX-X-: all down
	(Now have *-------)
	XXX-XXX-: if any u, invert it, ring
		if not, invert ***.---.
	(Now have ****----)
	XXXXXX--: four u, invert them, ring
		four d, invert them, ring
		else, invert *-**-*..
	XXXXX-X-: invert *-*-*.*., ring

W=9 H=2 - impossible; 9/3 is impossible
W=9 H=3 - impossible; 9/4 is impossible
W=9 H=4 - impossible; 9/5 is impossible
W=9 H=5 - impossible, basically same proof as W=6 H=3, but with
*--*----- as the relevant pattern instead of *-*---.
14 patterns:
XXXXX---- XXXX-X--- XXXX--X-- XXXX---X- XXX-XX--- XXX-X-X-- XXX-X--X-
XXX--XX-- XXX--X-X- XXX---XX- XX-XX-X-- XX-XX--X- XX-X-XX-- XX-X-X-X-

W=9 H=6 - unsure
10 patterns: XXXXXX--- XXXXX-X-- XXXXX--X- XXXX-XX-- XXXX-X-X- XXXX--XX- XXX-XXX-- XXX-XX-X- XXX-X-XX- XX-XX-XX-

W=9 H=7 - possible
4 patterns: XXXXXXX-- XXXXXX-X- XXXXX-XX- XXXX-XXX-
	XXXXXXX--: all down
	XXXXXX-X-: all down
	(Now have *--------)
	XXXXXX-X-: one u, invert it, ring
		no u: invert -------*-
	(Now have **-------)
	XXXX-XXX-: two u: invert them, ring
		*---.---.: invert *-*-.*--.
		---*.---.: invert -*-*.--*.
		----.*--.: invert -*--.*-*.
		----.--*.: invert --*-.*-*.
	(Now have *--*--*--)
	XXXXX-XX-: if three u, invert them, ring
		else invert all d, ring

W=10 H=2 - impossible; 10/3 is impossible
W=10 H=3 - impossible; 10/4 is impossible
W=10 H=4 - impossible; 10/5 is impossible
W=10 H=5 - impossible, basically same proof as W=6 H=3
26 patterns:
XXXXX----- XXXX-X---- XXXX--X--- XXXX---X-- XXXX----X- XXX-XX---- XXX-X-X---
XXX-X--X-- XXX-X---X- XXX--XX--- XXX--X-X-- XXX--X--X- XXX---XX-- XXX---X-X-
XXX----XX- XX-XX-X--- XX-XX--X-- XX-XX---X- XX-X-XX--- XX-X-X-X-- XX-X-X--X-
XX-X--XX-- XX-X--X-X- XX--XX--X- XX--X-X-X- X-X-X-X-X-

W=10 H=6 - unsure
22 patterns:
XXXXXX---- XXXXX-X--- XXXXX--X-- XXXXX---X- XXXX-XX--- XXXX-X-X-- XXXX-X--X- XXXX--XX-- XXXX--X-X- XXXX---XX- XXX-XXX---
XXX-XX-X-- XXX-XX--X- XXX-X-XX-- XXX-X-X-X- XXX-X--XX- XXX--XXX-- XXX--XX-X- XXX--X-XX- XX-XX-XX-- XX-XX-X-X- XX-X-XX-X-

W=10 H=7 - unsure
12 patterns:
XXXXXXX--- XXXXXX-X-- XXXXXX--X- XXXXX-XX-- XXXXX-X-X- XXXXX--XX-
XXXX-XXX-- XXXX-XX-X- XXXX-X-XX- XXXX--XXX- XXX-XXX-X- XXX-XX-XX-

W=10 H=8 - possible, generalization of W=6 H=4
	XXXXXXXX--: all down
	XXXXXXX-X-: all down
	(Now have *-------)
	XXXX-XXXX-: if any u, invert it, ring
		if not, invert ****.----.
	(Now have *****-----)
	XXXXXXXX--: five u, invert them, ring
		five d, invert them, ring
		else, invert *-*--*-*..
	XXXXX-X-: invert *-*-*.*., ring

Also interesting is what patterns are at depth 1, that is, what sets
are one step from a guaranteed bell ring.  Obviously, this makes sense
for only the solvable problems.  All such sets so far are of size one.
These all seem to be related to the factors of W, reinforcing the idea
that W=prime>3 2<=H<=W-2 is impossible.

W=4 H=2: *-*-
W=6 H=4: **-**- *-*-*- *--*--
W=8 H=4: *-*-*-
W=8 H=5: *-*-*-
W=8 H=6: ***-***- *--*-- *-*-*- **--**--
W=9 H=7: **-**-**- *--*--*--