dailysudoku.com

[Marty R.]

I've tried this about five times, each time backing myself into a corner with a duplicate number. I'm ready for a new eraser. If someone's willing to take a shot at it and let me know what techniques you used to solve it, that would give me more confidence to take a sixth shot at it. Thanks.

(Do others have this problem of not being able to let go of a puzzle?)

[keith]

Marty,

Using usual techniques and one XY-wing, I get to this point:

[David Bryant]

Hi, Marty!

If I start a puzzle I always finish it. I might have to put it aside for a day or so, and take a fresh crack at it, but I don't give up on these things!

Presumably you got this far without any trouble, because all the steps I had to make up to this point were pretty simple.

[TKiel]

Marty R,

Right off the bat I placed 12 naked and hidden singles. Then, and I must admit I didn't notice this until my second time through in solving it, I spotted a naked quad in box 5 (needless to say I didn't spot the hidden triple in the same box, until I'd made the exclusions from the quad and it became a naked triple). Then there is an x-wing that makes exclusions from 3 cells and that allows the placement of another single. After that I found locked candidates in box 5 that allowed an exclusion from one cell. That exclusion allowed the placement of two cells in a different digit.

After that, get out your colours. The remainder of the puzzle solves mainly with colouring, mostly one chain but sometimes two, except there is a hidden pair in box 2, that you will find after a 5 has been placed in r2c2. I'm tried not to give away too much so I tried to be as cryptic as possible in the description of what I did. If you want more detail, give a shout.

[keith]

Marty,

This Unique Rectangle is a little different from the one discussed yesterday, so let me expand.

In a solved puzzle, the pattern

[David Bryant]

Hi, Marty! I hope you're getting enough help on this puzzle.

In comparing the position I had reached before I started using colors on this puzzle against the first position Keith posted I noticed something pretty interesting. The only difference is that Keith had a "3" in r7c1 and in r5c3, which I had not yet solved for. Here's the position as I posted it, with the candidate lists filled in.

[keith]

Marty,

Where did you find this puzzle?

David, thank you for the compliment(s). I thought this coloring is really cool. Otherwise, how would you solve the puzzle?

Keith

[David Bryant]

[Marty R.]

[keith]

Marty,

I talked about coloring in the thread "Nightmare of Feb 28". Here is the essential method, with some further explanation:

How does (simple) coloring work? For any value of a possibility (the "coloring value"), look in each element (row, column, block) of the puzzle to find where the possibility occurs exactly twice. Draw a line connecting the two squares.

(Actually, I just use a separate piece of squared paper with a 9x9 empty grid to do this.)

When you are done, you may have a collection of "graphs", networks of lines that connect pairs of possibilities. Now, you can use two colors (say, red and green) to alternately label each square on the graph. Either the red squares have the possible value, or the green squares do.

Another convention is to label alternate nodes of the graph with upper and lower case letters. "A" and "a" for the first graph, "B" and "b" for the second, etc.

There are two implications to look for:

1. Two squares in the same element (row, column or block) have the same label, say "a". Then, the squares labelled "A" must be the coloring value.

2. Squares which are neighbors of both of any two squares labelled "a" and "A" cannot have the coloring value. This is what applies here.

So, coloring is a completely explainable technique. You do not need some sixth sense or special ability to spot patterns. In fact, the coloring process, of connecting the "conjugate" squares, will also help you to find swordfish, etc.

You can find more by searching on "sudoku coloring" on Google. Be sure to also search on the British spelling, "sudoku colouring". Personally, I found the explanation on the SadMan site to be the most helpful.

Best wishes,

Keith

[TKiel]

Marty R,

One of the advantages of explaining a technique to someone else is it often helps to crystalize things in ones own mind, so it actually benefits both parties. Thanks for the kind words.

In regard to your question about colouring, the cells really select themselves. As I understand how it all came about, angus johnson (www.angusj.com) first called the technique colouring (which he acknowledges he did not first propose) because the puzzle generator he wrote allowed assigning different colours to particular cells, for purposes of identification. He used different colours to indicate the relationship between the cells and the technique became known as colouring.

Simple colouring refers to labeling cells that are conjugates of each other. Conjugate means they are the only two cells in a group (row, column or box) that have a certain digit as a candidate. Conjugate cells have a special relationship: not only is it a fact that if one is true, the other is false, it is also a fact that if one if false then the other is true. This is immensely powerful. If cell 1 & cell 2 are conjugates, if I prove cell 1 true, then I know that cell 2 is false. It's also true that if I prove cell 2 false, then I know that cell 1 is true. The end result is the same (cell 1 = true, cell 2 = false) but I have two ways to go about proving that, either by eliminating the possibility of cell 2 = true or by actually proving cell 1 = true.

Labeling one cell with one color and a conjugate cell with a different color implies nothing about which is those cells is true or false. It only allows the solver to see the relationship. When cells can be linked with alternating colours in such a manner it is called a conjugate chain or just a chain. All the cells marked with blue will end up being true or all the cells marked with green will end up being true; we just don't know which. When a cell that is not part of the chain shares a group (row, column or box) with a cell marked with each colour, we know that cell cannot possibly be true.

This grid is from the puzzle you posted at the top of this thread. Instead of colours, I've use (A) and (a) to indicate states of oppositeness.