If anybody else has got any good tips on solving without pencilmarks, please post them here and we can try to make a little collection of these techniques.
First of all, whenever I say that I solve without pencilmarks, peoples first reaction is: "then you must have a really good memory". My answer is yes, but I hardly ever use it. The most important thing I've learned so far is to choose what information to memorize. There is simply so much information to be found in any grid, that one can't remember it all. Most of this information is useless. When I started doing Sudoku I often found myself trying to memorize possible values for cells, drawing a mental pencilmarkgrid. When I finally realised that this is pointless, my solving improved significantly. Most puzzles can be solved exclusively with naked or hidden singles, then what's the point in knowing that a cell can hold numbers 1,2,6 and 7? In fact, all numbers are solved by naked or hidden singles, all other techniques only reduce possibilities. So what I’m interrested in is these reduced possibilities. Instead of memorizing possible candidates, I memorize impossible candidates. As most of the impossibles can be read from the column, row and box of each cell, there isn’t really that much to remember. Only when I make a reduction through some other technique, like x-wing, uniqueness etc., I memorize the reduced candidates. Apart from this I memorize locations of pairs and triplets, but the rest of the information about each cell I put aside as soon that I can see that it doesn’t help me at the moment.
I guess most of you use pencilmarks to spot singles, so the most important thing to improve if you want to solve without pencilmarks is not your memory, but your ability to spot singles. The hidden singles are usually easy to spot, just picture horisontal and vertical lines going through all instances of a particular number and find the only empty square. Naked are harder to find. Going through the whole puzzle cell by cell is very timeconsuming. I rather look for them one row or column at a time. I start with the rows/columns that have most numbers already, then I count from 1 to 9, leaving out all the numbers already in the row. Example:
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*-----------*
|..1|..2|35.|
|...|...|2..|
|.8.|4..|.91|
|---+---+---|
|5.9|31.|4..|
|...|7.4|...|
|..7|.25|8.3|
|---+---+---|
|86.|..9|.3.|
|..3|...|...|
|.74|1..|5..|
*-----------*
If I wanted to find naked singles in this puzzle, I would start with row 4 that already has 5 numbers, then count: “. 2 . . . 6 7 8 .” These are the 4 numbers that go in the four empty cells, so now I compare each cell against these numbers. R4c2 has 6, 7 and 8 in the same column, so there’s a naked single (2). This trick is also efficient when looking for hidden singles in rows or columns (try it on row 6). If you systematically go through all rows like this you will find all naked singles in no time.
So when do I need the good memory? That’s only when I can’t find anything else and start “trailing”. Trailing is a common term I like to use for any technique that involves reading many steps ahead. The different trails I’m looking for are trails that lead to contradictions, double solutions or forcing chains (which obviously involves at least two trails). Almost any puzzle can be solved with relatively short (<5 steps) trails, only the most evil puzzles require longer. Reading many steps ahead requires a lot of practise, but it’s not in any way impossible to learn. You may start your practise on easy puzzles that you can effectively solve without marks already. Instead of writing in an obvious number, try to find one or two other numbers solved by your original number and then write them in all at a time. Then you can gradually extend the trail. I used to have problems reading more than 3 steps ahead, now I usually solve all puzzles up to moderate level in my local newspaper one number at a time - don’t write down anything until I solved all instances of the number. This might seem hard at first, but with proper training anybody can run a marathon.
Another question I often get is ”how can you use advanced techniques like coloring or XYWhatever-wing without pencilmarks?” The answer is very simple, I don’t use them. Techniques like these are in my opinion not at all advanced, but limitations of the real techniques behind them. They are limitations in the sence that they actually apply different trailing techniques, but only on very special patterns that somebody has predefined for you as they can be seen in a pencilmarkgrid. When programming your solver software this might be neccessary, but as a human solver you should be able to see the logic in any situation.
Take for example “multiple colors”. When I first got Simple Sudoku I opened a “multiple colors”-example to see what it was about. First I solved it in my way, then I checked how SS would have done it. I found that it colored all cells with candidate 2 in five different colors, then removed some candidates of number 2. I could see that all these candidates shared an unit with a 2 that I had solved with a short forcing chain. I then checked all the other examples of multiple colors and found that in every case a cell connecting all removed candidates could be solved with a simple forcing chain. So apparently this is a technique that you can use instead of a forcing chain if it considers only one candidate number, if all instances form a specific pattern and if you have a good set of crayons. Well, then I prefer the forcing chain as it applies to endless situations where the crayons won’t help you. Guess you want to see the example:
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*-----------*
|78.|3..|.1.|
|615|.8.|.32|
|3.9|.1.|8..|
|---+---+---|
|1.8|..3|.6.|
|.73|462|1..|
|.6.|1.8|7.3|
|---+---+---|
|..1|.3.|6..|
|85.|.4.|3.1|
|.3.|..1|..5|
*-----------*
I used this forcing chain:
if r1c3=2 => r9c5=2 => r4c7=2
if r3c2=2 => r6c1 or r6c3=2 =>r4c7=2
Simple sudokus advanced spread of colors across the grid ended up in removing candidate 2 from r4c2, r6c8 and r9c7, which I obviously also could do after solving r4c7 with the forcing chain.
My last tip before getting on with the actual puzzles is this: UNIQUENESS!! Uniqueness patterns are everywhere and they often provide a very easy solution to extreme puzzles. The best thing with them is that they are very easy to spot. Again, don’t limit yourself to the reductions explained in different forums, be creative. My approach is that whenever I see two corners of a uniqueness square in place, I put in a third corner and read ahead to see if it gives me the forth. Actually, whenever I could make any move that results in three corners of the square in place I read ahead to see if the forth falls in place. It often does, so these are very good spots to start your trails from if you're stuck.
Now let’s have a look at some extreme puzzles. I recommend that you copy them into some sudoku program (without pencilmarks) and follow as we go along. To help you follow without pencilmarks I’ll use these abbreviations on how I solve each number:
Ce = only possible number that can go in the cell
Co = only possible cell for number in column
R = only possible cell for number in row
B = only possible cell for number in box
First an extreme from Simple Sudoku:
Puzzle 1
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*-----------*
|.2.|1.8|..5|
|1..|.76|.3.|
|7..|4..|...|
|---+---+---|
|..7|...|2..|
|48.|...|.59|
|..2|...|6..|
|---+---+---|
|...|..9|..1|
|.5.|74.|..6|
|6..|8.1|.7.|
*-----------*
First the obvious ones:
r7c2=7 (B)
r1c7=7 (B)
r6c9=7 (B)
r5c6=7 (B)
r8c3=1 (B)
I can’t see anymore obvious cells by drawing lines, but I can see two pairs: r78c1={2,8} (R) and r23c3={5,8} (B). Next I start checking rows, starting with row 1. Two pairs there also: c1&5={3,9} and c3&8={4,6}. This is enough to solve the next number:
[edit: typo there, the first pair should read r78c1={2,8} (Co) = the two numbers cannot go anywhere else in the column.]
r9c3=9 (Co)
This gives me another pair: r7c3 and r9c2={3,4} – so far 5 pairs to remember. Now I see a very obvious trail:
if r1c1=9 => r3c2=3 (B) => r9c2=4 (Co) => r1c3=4 (B) => nowhere to place 6 in box 1. This gives me:
r1c1=3 (Ce)
r1c5=9 (Ce)
At this point I can tell that 5 cannot go in r4c4 or r6c4 ({9,5} uniqueness square with c1), always worth to notice.
I decide to check the other pair of row 1 and find:
if r1c3=6 => r1c8=4 => r4c9=4 (B) and
if r1c3=6 => r2c2=4 => r9c2=3 => nowhere to place 3 in column 9. This gives me:
r1c3=4 (Ce)
r1c8=6 (Ce)
r3c2=6 (B)
r5c3=6 (B)
r7c3=3 (Ce)
r9c2=4 (B)
r2c2=9 (B)
Now I have:
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*-----------*
|324|198|765|
|19.|.76|.3.|
|76.|4..|...|
|---+---+---|
|..7|...|2..|
|486|..7|.59|
|..2|...|6.7|
|---+---+---|
|.73|..9|..1|
|.51|74.|..6|
|649|8.1|.7.|
*-----------*
Still can’t see any obvious so I have a look at column 9, missing numbers 2,3,4 and 8. This gives me an idea that I have to check out and this is what I see:
if r3c5 or r3c6=2 =>r2c9=2 (B) and
if r2c4=2 => r2c3=5 (R) => r3c3=8 => r3c9=2 (Ce)
Anyway, r9c9 cannot be 2. This gives me a 2 in either r7c8 or r8c8 that not only forms an x-wing with c1 but also lets me exclude 8 from both squares (uniqueness). Thanks to this I can go on:
r8c6=3 (Ce)
r9c5=2 (B)
r5c4=2 (B)
r3c6=2 (B)
r2c9=2 (B)
and so on... rest of the puzzle is only singles so I guess you can do it yourself. There the puzzle was solved with a few short trails, max 4 steps. Could you follow without pencilmarks? If you could, then you can also do this yourself. If you try enough short trails you will find lots of reductions like the ones I made here. As you could see the only neccessary memorizing I did was the 5 pairs, that soon could be forgotten as they were solved (or made obvious by filling in the rest of the boxes), and then i memorized max 4 numbers at a time while trailing.
Let’s have a look at one more puzzle, this is a Solo extreme, that Simple Sudoku cannot provide a logical solution for. Let’s find out how hard it really is:
Puzzle 2
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*-----------*
|1..|.32|5..|
|5..|...|.92|
|7..|5..|4..|
|---+---+---|
|...|.2.|71.|
|...|4.8|...|
|.21|.5.|...|
|---+---+---|
|..8|..5|..7|
|31.|...|..6|
|..7|84.|..9|
*-----------*
First the singles:
r3c9=1 (Co)
r5c2=7 (B)
r1c8=7 (B)
r1c9=8 (Ce)
r2c6=4 (B)
r4c1=8 (B)
r3c3=2 (B)
r5c5=1 (B)
r2c4=1 (B)
r2c5=7 (B)
r2c2=8 (R)
r3c5=8 (B)
r9c6=1 (B)
r7c7=1 (B)
r7c5=6 (R)
r8c5=9 (Ce)
r7c4=3 (B)
r8c4=2(B)
r8c6=7 (Ce)
r6c4=7 (B)
r8c7=8 (Ce)
r6c8=8 (B)
This is the situation:
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*-----------*
|1..|.32|578|
|58.|174|.92|
|7.2|58.|4.1|
|---+---+---|
|8..|.2.|71.|
|.7.|418|...|
|.21|75.|.8.|
|---+---+---|
|..8|365|1.7|
|31.|297|8.6|
|..7|841|..9|
*-----------*
At this point I can’t see any more obvious cells, so I start reading through rows and columns. In column 1 I react on r5c1 that has to hold 6 or 9. This tells me that if r5c7=9 => r5c1=6 (Ce) => r6c7=6 (B). That’s three corners of the uniqueness square so I decide to follow the trail a bit further. And, not to my suprise, I find three steps ahead:
if r5c7=9 => r5c1=6 (Ce) => r6c7=6 (B) => r3c8=6 (B) => r3c6=9 (Ce) => r6c1=9 (R) => Double solution. I can remove that option and go on:
r6c7=9 (B)
r5c1=9 (Ce)
r7c2=9 (B)
r1c3=9 (B)
r3c6=9 (B)
r4c4=9 (B)
r1c4=6 (Ce)
r1c2=4 (Ce)
Current situation:
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*-----------*
|149|632|578|
|58.|174|.92|
|7.2|589|4.1|
|---+---+---|
|8..|92.|71.|
|97.|418|...|
|.21|75.|98.|
|---+---+---|
|.98|365|1.7|
|31.|297|8.6|
|..7|841|..9|
*-----------*
Now this trail immediately jumps into my face:
if r4c2=3 => r6c6=3 (B) => r6c1=6 (R) => no way to fit numbers 3 and 6 into column 3. I can remove that option and continue:
r3c2=3 (Co)
r2c3=6 (Ce)
...you may fill in the remaining singles. That wasn’t so hard, was it? This time I didn’t actually memorize anything except the two short trails.
I hope I have showed by this that memory isn’t the biggest issue when solving harder puzzles without pencilmarks, but learning to see patterns and short trails. Don’t focus on trails you’ve read about in technique guides, because (a) you won’t see them (they are defined by possible candidates) and (b) they are very limited. With pure logic and the ability to read 5 steps ahead you can solve almost any puzzle this way.
regards, RW
PS. Seems my trails have a lot in common with what you call nice loops. If so, feel free to translate them into the "correct" language that I don't really master yet.
