## Shortz’ Ferocious Hall of Mirrors

This post recounts the solving of Beware 190, the review puzzle from Will Shortz’ collection book Ferocious Sudoku which has multiple solutions.  On another multiple solution puzzle, Frank Longo’s  Absolutely Nasty Level 4 puzzle 61(Review Snaps II 3/12/13), I used coloring to unravel the tangle of solutions. Here ALS aided nice loops aid this process.

Let’s start this time with 7 solutions.  The good side of this result is that it is so evident and verifiable that 7 solutions exist.

With my puzzle template, it’s easy to display up to nine solutions on one grid. Each solution is displayed in its own keypad  cell position. Here, the solution given by Will Shortz in Ferocious is in the upper left corner, plus the values in orange and white script. These values are now known to  belong in all solutions

Now let’s consider how to get there.  The eighth position in the C box represent the blue side of a blue green cluster.  While green was proved right, seven times in fact, blue was not proved wrong.  That happens with multiple solutions. I follow up on green in this post.

This all begins quite ordinarily. The box marking 1-D trace is not convoluted.  The line marking trace is even simpler, because the accumulation of candidates is all that happens:

In the resulting grid, clouds of candidate numbers hover over about half of the boxes, while other units are relatively clear and have bv and other ALS.  Too many or too few candidates for SdC, and poor alignments for APE.  I do find one 379-wing.

The bv map reveals that the bv are also poorly coordinated for XY chains.

Moving to the X-panel, I encounter another case of too many or too few candidates, except for a productive finned swordfish. These results suggest an overabundance of candidates.

Coloring produces two very limited clusters with a bridge, namely :

not(red and green)=> orange or blue.

Adding AIC hinges doesn’t help, but I do manage a simple AIC chain that removes the wrong 4-candidate, not the one that would have created an ER 346-wing.

Next is the enumeration of ALS, for toxic sets and AIC nodes. No removals, and the two  ALS aided AIC  I found were nice loops that defined the same two alternatives.

In this case, both the left and right nice loops declare that either

A:(E 9grp and 2 in r4c9 and SE2grp)

or

B:(SE9grp and 2 in r7c9 and 2 in r4c8)

is true.

Alternative A => blue and orange, but alternative B makes no decision.

I have never called upon nice loop alternatives before, but it’s obvious when you think about it, that in a long nice loop, one set of nodes, even or odd, is true.  We can add it to color clusters as a means of defining alternatives, especially when we are getting suspicious of multiple solutions.

Moving on to LPO,  interactions between patterns produce no solutions, but I do get some progress. The 7-panel yields two blue patterns, and a single green one.  The dashed blue pattern ends on a red candidate, and therefore cannot be alternative A above, since A implies blue and orange.

The single green pattern requires an orange candidate, effectively merging green and orange.  B is either green/orange, or  blue/red.

With all techniques I have blogged about apparently confounded, I turn to the successive coloring breakdown I did with Longo’s bad boy. The green&orange alternative resulted in the 7 solutions displayed earlier

Here I start off the breakdown with alternative B and blue&red, and display the resulting four more solutions.  This leaves the completion of this breakdown, and alternative A for you to explore if you like, but there’s not going to be a checkpoints.  I’ve seen enough.

To begin the B&blue&red breakdown, we add a gold/green cluster on the candidates.

Choosing gold, elimination of the green candidates brings a few removals, and we add a purple/copper cluster.

We add a grey/sunset cluster to purple, for two solutions.  Then a tan/yellow cluster to copper, for two more.

These four solutions are displayed in the four corners.

In closing, an observation:

Guessers will hit a solution quite easily, because the lower left third of the cells seem to accommodate to anything they see.  Most will not check the solution. So they think they have solved this puzzle, but they haven’t.  Or of those that check the solution, most will not verify that their different solution is a solution.  They will think they made a mistake, but they didn’t.

Next time, we review another “Beware! Very Challenging” Will Shortz collection. You may want to be sure this review was not an unlucky fluke.

As is our custom, here is a preview challenge to go as far as you can with a review puzzle, for checkpoints next post.  Its Beware 131 from Will Shortz’ Trickiest Sudoku.