Web Paint-by-Number Forum
Topic #110: Diagonal Logic
By Eric Francis (airdrik)

#1: Eric Francis (airdrik) on Aug 5, 2008

I've figured out another form of logic, which I will term: diagonal logic. It is presented when there is a set of clues which can be deduced to be linked together to from a diagonal line, for instance:
``` 1121211
3
3
3
```

By itself, the diagonal has two solutions (usually lines going diagonally in different directions)

``` 1121211  1121211
3ooo.... 3....ooo
3..ooo.. 3..ooo..
3....ooo 3ooo....
```
To produce a unique solution, something must interrupt the diagonal somewhere:
```  1
1121211
13o...ooo
3..ooo..
3ooo....
```
There are exceptions where one (or more) solution may not be a completely connected diagonal line:
``` 12222  12222  12222
2oo... 2.oo.. 2...oo
2.oo.. 2.oo.. 2...oo
2..oo. 2...oo 2.oo..
2...oo 2...oo 2.oo..
1....o 1o.... 1o....
```
(from what I can tell, the exceptions are only those where the clues are all 2s where there is an ending 1 in both the row clues and the column clues)
#2: Jan Wolter (jan) on Sep 3, 2008
I took the liberty of inserting a little HTML into your posting to make the pictures look as you typed them in.

I just ran across puzzle #3415 which contains an instance of this. After lots of work, you are left with a rectangle with just one clue in each row and column. It is easy to see that it is possible to solve it by a diagonal, but it is not obvious at all to me that that is the only solution.

As was demonstrated here, sometimes things that can be solved by a diagonal can have many other solutions, and sometimes they have a unique solution. I don't really know how to tell them apart except by trial and error.

I suppose if there was a simple rule for identifying when a diagonal solution works then this could be treated as valid logic solution trick, but I don't know what that rule is, or even if such a rule exists.
#3: Eric Francis (airdrik) on Sep 4, 2008
Thank you for the html.

As for determining if there is a unique solution; what I have figured, so far, is that the only case where there are many solutions is where each set of clues has one 'tail' ending in 1 followed by all 2s (like the second example). If, on the other hand, both tails on one set of clues end in 1, or there are other clues in the mix, then there will be a unique solution.

Also, symmetry applies: if the clues are symmetrical, there will be 2 solutions, otherwise there will be one solution (with the exception when there are many solutions).

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