motris wrote:Sure. I guess I'd simplify my comments by saying there are two axes of difficulty with a sudoku. They can have easy or difficult (or non-existent) logical steps. They can have a broad or narrow solving path (meaning fewer places to make the next deduction). These have a low score on the type of step axis and a very high score on the solving path axis. For those who don't track where they have and haven't looked as well, these puzzles can have the feeling of "I'm not making any progress" or "It's coming in fits and starts".
I agree. I've tested these four grids in my own sudoku software, which grades puzzles on all of these factors - in particular it looks at how many different deductions can be made simultaneously and knows that if there are lots then it is a much easier puzzle than if there is one. Furthermore it knows that spotting a "hard" logic move when there are very few options left in the puzzle is pretty trivial compared to spotting it as the very first move in the puzzle.
Based on testing these puzzles I've rated them (for Sudoku 5,6,7,8 in the test) as 3801, 2579, 3336 and 2369 respectively, where lower is easier. For comparison, a very easy sudoku can rate as low as 30 and in this case there will be many easy placements simultaneously available. A very hard sudoku, with a few difficult sets or x-wings that need to be spotted early on in a mess of pencilmarks will score 7000+. The hardest sudoku I've made scores just over 10000. However this still makes the ones in this test pretty tough sudoku, for those of us not blessed with your number-spotting talents.
Without pasting in a curve of value distribution, if I generate puzzles completely randomly I get puzzles as hard as the first one around 1 in 400 times. So I'd say it was not surprising most people found it tough.
In particular the first one (puzzle 5) requires lots of naked singles fairly early on in the solve, and in each case there is only one spot to make progress. Based on analysing 100,000s of plays on puzzlemix.com I am pretty certain that naked singles for the average solver can be very hard to spot indeed - although ironically they show up trivially if you write in all the pencilmark candidates.
Here's my analysis for each puzzle. The number on the left shows the number of empty squares and the logic required at each stage to progress. If the next number is 1 less then there was only 1 place to apply this logic, so it might have been very hard to spot (certainly you have a lower chance of stumbling on the next 'move' quickly). Obviously this is not a full tree so there may be alternative harder moves at a particular point as an alternative (and you may disagree on my grading of what is harder), but this gives a good guide to overall difficulty:
Puzzle 5: (lots of only-one-place-to-progress naked singles early on - tough)
55=uniqueInRegion 53=uniqueInRegion 52=nakedSingle 51=intersection 51=intersection 51=nakedSingle 50=nakedSingle 49=intersection 49=intersection 49=nakedSingle 48=uniqueInRegion 47=nakedSingle 46=uniqueInRegion 45=nakedSingle 44=nakedSingle 43=nakedSingle 42=nakedSingle 41=nakedSingle 40=uniqueInRegion 38=uniqueInRegion 37=nakedPair 37=uniqueInRegion 35=uniqueInRegion 33=uniqueInRegion 30=uniqueInRegion 28=uniqueInRegion 26=uniqueInRegion 23=uniqueInRegion 21=uniqueInRegion 19=uniqueInRegion 17=uniqueInRegion 15=uniqueInRegion 11=uniqueInRegion 7=uniqueInRegion 3=uniqueInRegion
Puzzle 6: (quite a bit easier, though still a narrow solving path)
56=uniqueInRegion 52=uniqueInRegion 51=nakedPair 51=intersection 51=nakedSingle 50=uniqueInRegion 49=intersection 49=nakedPair 49=uniqueInRegion 48=uniqueInRegion 46=uniqueInRegion 44=uniqueInRegion 43=uniqueInRegion 41=uniqueInRegion 38=uniqueInRegion 35=intersection 35=intersection 35=uniqueInRegion 34=uniqueInRegion 33=uniqueInRegion 32=nakedSingle 31=uniqueInRegion 30=nakedSingle 29=uniqueInRegion 28=uniqueInRegion 24=uniqueInRegion 17=uniqueInRegion 6=uniqueInRegion
Puzzle 7: (more only one-place-to-go naked singles early on make it tough, followed by various intersections and naked pairs needed, also early on)
54=uniqueInRegion 51=nakedSingle 50=nakedSingle 49=nakedSingle 48=intersection 48=intersection 48=nakedSingle 47=nakedSingle 46=intersection 46=intersection 46=intersection 46=nakedPair 46=intersection 46=nakedPair 46=uniqueInRegion 45=uniqueInRegion 44=uniqueInRegion 43=uniqueInRegion 41=uniqueInRegion 38=uniqueInRegion 37=uniqueInRegion 35=uniqueInRegion 33=uniqueInRegion 31=uniqueInRegion 28=uniqueInRegion 21=uniqueInRegion 14=uniqueInRegion 12=uniqueInRegion 9=uniqueInRegion 4=uniqueInRegion
Puzzle 8: (a few only-one-place naked singles, but relatively easy and ironically the only one I couldn't solve...)
56=uniqueInRegion 54=uniqueInRegion 52=uniqueInRegion 51=nakedSingle 50=uniqueInRegion 48=uniqueInRegion 47=nakedSingle 46=intersection 46=intersection 46=intersection 46=intersection 46=intersection 46=nakedPair 46=uniqueInRegion 45=uniqueInRegion 44=uniqueInRegion
43=uniqueInRegion 40=uniqueInRegion 39=intersection 39=uniqueInRegion 38=uniqueInRegion 36=uniqueInRegion 35=uniqueInRegion 32=uniqueInRegion 26=uniqueInRegion 15=uniqueInRegion 5=uniqueInRegion
I'm not really sure why the second two were worth more points. In fact I only just noticed that!