I nearly forgot to mention how easy it is to construct an invalid Jigsaw layout. All you need is 3 regions like this. A A A A A A C B B B C C C B B B Assume the grid size is 6. So the A in column 2 has to match the C value in the corner. If that's 3, say, then you can't put a 3 in region B. That pattern can be extended to any grid size. Note that region C canby Mathimagics - Sudoku Xtra Discussion Forum
QuoteJim 8x8 invalid layouts can take up to a minute to detect, and 9x9's up to an hour! This turns out to be wildly inaccurate. I now realise I haven't actually detected an invalid order-9 Jigsaw layout, because all the ones whose testing I had abandoned (in one case after 3 hours) have since turned out to be valid! And with oodles of solutions, too. The reason is that the larger the griby Mathimagics - Sudoku Xtra Discussion Forum
I should add that the only available results concerning the complexity of the decision problem for grid layouts are related to the regular layout, in which all regions have N cells. That's the Gerechte Design decision problem. Now that I have a generalised layout system in which regions can be of arbitrary size, I seem to have ventured into the unknown. Theoretically regions smaller or largerby Mathimagics - Sudoku Xtra Discussion Forum
Unfortunately, all that is known is that the decision problem (is a layout solvable?) is difficult, and just as difficult as the search problem (finding a solution). In complexity theory terms, both problems are NP-hard (and the decision problem is also NP-complete). And in practice, well, testing a layout is effectively the same problem as finding a solution. My GDL (GD layout) test functiby Mathimagics - Sudoku Xtra Discussion Forum
A regular Jigsaw layout is the one that puzzlers are all used to - every region has N cells. I have to use "regular" to distinguish them from those irregular layouts that I've now started to use. Regular puzzlers like the good folk here at SudokuXtra see lots of regular Jigsaw layouts, but have you ever wondered whether all layouts are valid? In other words are there impossible layby Mathimagics - Sudoku Xtra Discussion Forum
I'm very happy to hear you like these puzzles. I enjoyed solving them myself so I am not surprised you liked them. Some clues for the cryptics: Draw up a yes/no grid like those used in Logic Puzzles, with the letters along the top and numbers 0 to N-1 down the side. This will help you eliminate potential letter/value combinations - crossing out those combinations you know can't work. Theby Mathimagics - Sudoku Xtra Discussion Forum
OK, I've got the generalised VSR system working, and have attached two samples, with grid size N = 7 and 8. Both have nice symmeteric layouts using 3x3 regions, and this definitely adds an interesting twist to the solving process. With N=7 a region with 9 cells has two values occurring twice, or one value occurring 3 times. For N=8 a 9-cell region has just the one value occurring twice.by Mathimagics - Sudoku Xtra Discussion Forum
Thanks for pointing me to that, I hadn't considered the possibility of regions with more than N cells (N being the grid size). I looked at the link you provided and I can see that "Surplus Sodoku" as defined must always involve a one-cell region on any size grid, since by that definition, you have N-1 regions of size N+1, which covers N2-1 cells which leaves one remaining. As youby Mathimagics - Sudoku Xtra Discussion Forum
Here as promised elsewhere is a Skyscraper Zero Killer puzzle - it's on an 8x8 grid, and has variously sized regions in which no digit can be repeated, and some (but not all) of the regions have a Killer sum indicated. Enjoy - although I should warn that this one is reasonably challenging! Hmm, now that I have four distinct SSZ puzzle formats (regular, VSR, Killer, Cryptic), I might haveby Mathimagics - Sudoku Xtra Discussion Forum
I must have missed the Killer Sudoku 0-8 you mentioned, but I'd argue that it's certainly not facile - anything that requires you to adjust your thinking proceses is IMHO a good variation to the standard puzzle.by Mathimagics - Sudoku Xtra Discussion Forum
QuoteAlso, further down the track, a new application for Zero - I think it will make things very interesting for Killer Sudoku (and Outside Sum). Oh dear, and this from a man who is allegedly a Computational Scientist! Zero makes no difference to the complexity of a straight Killer puzzle, since there is obviously a 1-to-1 correspondence between sums of digits in range [1 to N] and range [0by Mathimagics - Sudoku Xtra Discussion Forum
One of the good things about using the value zero in number-place (Latin Square) puzzles is that we can then construct puzzles on a 10x10 grid and yet still require only one digit to be placed in each cell. But a 10x10 grid requires Sudoku(Jigsaw)-style regions to be defined, otherwise you need to have a very large number of "givens". The only "regular tiling" (by which Iby Mathimagics - Sudoku Xtra Discussion Forum
By the way, some future offerings I hope to provide involving Skyscraper Zero: Skyscraper Zero 10x10 with Jigsaw/Regions Samurai Skyscraper Sudoku Zero Samurai Star Skyscraper Sudoku Zero Also, further down the track, a new application for Zero - I think it will make things very interesting for Killer Sudoku (and Outside Sum).by Mathimagics - Sudoku Xtra Discussion Forum
Regarding the "givens", I forgot to mention them but what you presumed was just so! It's true that for regular (not Zero) Skyscraper Cryptic puzzles the 1 value is easily determined, it is extremely unlikely that more than one symbol could occur exactly once on each of the four sides. However, this does not mean that (the decryption problem for) regular SS Cryptics can't be haby Mathimagics - Sudoku Xtra Discussion Forum
Inspired by KrazyDad's "Krypto-Kakuro", I bring you Skyscraper puzzles with the added feature of using letters instead of digits for all clues. In the attached sampler you'll find four "Skyscraper Zero Cryptics", two of size 6 and two of size 7. Coincidentally, just as the value 1 is easiest to determine in Krypto-Kakuro, it is also the case with SS Cryptics, althoughby Mathimagics - Sudoku Xtra Discussion Forum
I love Samurai Star Skyscraper, it's usually the first puzzle I do in each issue. It should come as no surprise that I prefer tough puzzles!by Mathimagics - Sudoku Xtra Discussion Forum
Yes, that formula for generating even perfect numbers was known to Euclid (c300BC). Odd perfect numbers, on the other hand are strongly suspected of being non-existent, but nobody has yet been able to prove this. This is one of the great unsolved problems in number theory.by Mathimagics - Sudoku Xtra Discussion Forum
I excel at things trivial - one day I hope to master something useful!by Mathimagics - Sudoku Xtra Discussion Forum
The numbers in that last puzzle (Issue 13, p16), i.e. the prime numbers 2,3,5,7,13,17,19,31,61 are not perfect, but are prime numbers for which there is a corresponding perfect number. Perfect numbers are numbers like 6 and 28 which are the sum of all their proper divisors. So we have 6 = 1+2+3, and 28 = 1+2+4+7+14. Now, each prime p in Gareth's list has the property that the Mersenne numbeby Mathimagics - Sudoku Xtra Discussion Forum
Other than number and logic puzzles, I also like English Cryptic Crosswords (I like to do the ones in the Guardian). They seem to exercise a part of the brain that number puzzles don't, or at least that's my theory!by Mathimagics - Sudoku Xtra Discussion Forum
Thanks Nikki, I was a Univeristy maths tutor in a previous life, to be useful I had to be able to explain things Christine, well done! Now look again at the grid. A can't be 0? Why not? Look at the only "sum" with an A in it.by Mathimagics - Sudoku Xtra Discussion Forum
QuoteNikki I probably have too much free time and a slightly 'obsessive' mentality but I find usually if you keep working on something for long enough it eventually falls into place. Um, er, snap! We don't do "hobbies", we just pursue obessions!by Mathimagics - Sudoku Xtra Discussion Forum
I found it easier to draw the grid on a blank piece of paper in which I put digit values as they became known. A hiint for puzzle 2. The letters for values 1 and 2 are obvious, but also you can immediately determine which letter must be 0 (zero) by a process of elimination.by Mathimagics - Sudoku Xtra Discussion Forum
Hi Christine, Thanks for getting me interested in yet another puzzle type! I'll give you some clues to get you started on the first puzzle. Obviously C=1, and given a 4-digit sum is "IE", I must be 2 (it can't be 3 since E can't be zero). We are given F+B = E, and both F and B are more than 2, so E must be at least 7. But it can't be more than 7 because the top row sums toby Mathimagics - Sudoku Xtra Discussion Forum
OK, no problem in that case. The first of those additional rules is one that is almost implicit, but could certainly use some clarification. The second one, regarding the endpoint, certainly solves the uniqueness problems!by Mathimagics - Sudoku Xtra Discussion Forum
Poor old issue 12 has some further glitches, it seems - I have today discovered that 3 of the 6 Tiger in the Woods puzzles on p46 have 2 distinct solutions. Note that these puzzles were not created by Gareth, so it's not his fault! I discovered this by accident, having never seen this type of puzzle before today, but prompted by Nikki's cry for help on the subject, I decided to have a go. Myby Mathimagics - Sudoku Xtra Discussion Forum
Nikki, check out davmillar's reply to my post regarding the multiple-solutions here You'll be pleased to know there were some "missing" rules, although on closer inspection I'm not sure that they give rise to any radically different strategy. Note that the first of the two missing rules is the one I suggested above was probably true anyway - the path can't re-cross either of theby Mathimagics - Sudoku Xtra Discussion Forum
I had never heard of these puzzles until you mentioned them, then I found Deb Mohanty's puzzles on p46 of Issue 13. If it makes you feel better, I'm pretty good at most SudokuXtra puzzles, but I did find these puzzles very tricky indeed! For this puzzle it seems to me you definitely need an eraser, as I suspect a certain amount of trial-and-error is inevitable. Erasing for me is a pracby Mathimagics - Sudoku Xtra Discussion Forum
Gareth, I'm happy to see them put in the magazine - as for 0 vs "gap", well I'll leave that up to you. Will you print the challenger? If so, I'll email you the solution - unless you've cracked it already?by Mathimagics - Sudoku Xtra Discussion Forum
I really like Skyscraper (SS) puzzles, so I wrote some software to generate them. In doing so, the idea for SSZ puzzles occurred to me. In these puzzles you are placing values 0 to N-1 instead of 1 to N. A value of zero means no building at all, and this has a subtle effect on the visibility clues when they occur on the outer edges of the square. For example, say N=6. In a normal SS puby Mathimagics - Sudoku Xtra Discussion Forum
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