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Dominosa: some more solver thoughts.

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I've been playing this game a fair bit recently, and it's probably
time I jotted down some of the deductions I've been doing in my own
brain that the game doesn't know about. (Also I had an algorithmic
thought about the area-parity technique.)
This commit is contained in:
Simon Tatham
2018-09-21 08:55:21 +01:00
parent 55be8e50db
commit cd9c544230
1 changed files with 33 additions and 6 deletions
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39
dominosa.c
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@ -13,13 +13,17 @@
* * rule out a domino placement if it would divide an unfilled * * rule out a domino placement if it would divide an unfilled
* region such that at least one resulting region had an odd * region such that at least one resulting region had an odd
* area * area
* + use b.f.s. to determine the area of an unfilled region * + Tarjan's bridge-finding algorithm would be a way to find
* + a square is unfilled iff it has at least two possible * domino placements that split a connected region in two:
* placements, and two adjacent unfilled squares are part * form the graph whose vertices are unpaired squares and
* of the same region iff the domino placement joining * whose edges are potential (not placed but also not ruled
* them is possible * out) dominoes covering two of them, and any bridge in that
* graph is a candidate.
* + Then, finding any old spanning forest of the unfilled
* squares should be sufficient to determine the area parity
* of the region that any such placement would cut off.
* *
* * perhaps set analysis * * set analysis
* + look at all unclaimed squares containing a given number * + look at all unclaimed squares containing a given number
* + for each one, find the set of possible numbers that it * + for each one, find the set of possible numbers that it
* can connect to (i.e. each neighbouring tile such that * can connect to (i.e. each neighbouring tile such that
@ -37,6 +41,29 @@
* things out after finding the subset, we must be * things out after finding the subset, we must be
* careful that we don't rule out precisely the domino * careful that we don't rule out precisely the domino
* placement that was _included_ in our set! * placement that was _included_ in our set!
*
* * playing off the two ends of one potential domino, by
* considering the alternatives to that domino that each end
* might otherwise be part of.
* + if not playing this domino would require each end to be
* part of an identical domino, play it. (e.g. the middle of
* 5-4-4-5)
* + if not playing this domino would guarantee that the two
* ends between them used up all of some other square's
* choices, play it. (e.g. the middle of 2-3-3-1 if another 3
* cell can only link to a 2 or a 1)
*
* * identify 'forcing chains', in the sense of any path of cells
* each of which has only two possible dominoes to be part of,
* and each of those rules out one of the choices for the next
* cell. Such a chain has the property that either all the odd
* dominoes are placed, or all the even ones are placed; so if
* either set of those introduces a conflict (e.g. a dupe within
* the chain, or using up all of some other square's choices),
* then the whole set can be ruled out, and the other set played
* immediately.
* + this is of course a generalisation of the previous idea,
* which is simply a forcing chain of length 3.
*/ */
#include <stdio.h> #include <stdio.h>