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Boggle Sudoku

Boggle Sudoku

Safe shaking occurs when you envelop the board in loosely wrapped Plastic Wrap. Naturally, the idea is to have the displayed letters (in any orientation) occur in each row, column and block of 9 cells. The cartoon shows a failed attempt. As an added bonus, once you’ve come upon the solution, you can look for small words in the language of your choice occuring contiguously left (backward), or right (forward) or up or down or diagonally.

The game Boggle (TM) was invented by Allan Turoff and usually comes in a 4 x 4 grid. Larger Boggle games use a 5 x 5 grid. There is also a children’s version of Boggle as well as a travel version. See Wikipedia for details.

I’ve gotten some feedback during last week and I am happy to report that the Website for color sudoku puzzles, mentioned last February, 2007, which was called Brainfreeze Puzzles is no longer “frozen”. The site has been revamped; Philip Riley and Laura Taalman’s Color Sudoku book was published in mid-2007.

Look for the Published Puzzles link for free puzzle variations. There are also tutorials for the Sudoku variants. An original innovation is the Bold X variant, with 6 diagonals, 2 main diagonals and 4 “subprime” neighboring diagonals, which have non-repeating digits as well (just not all the numbers).

I’m glad the site is active again. It is a treasure trove of color and variants.

Another viewer simply named Maff, from the United Kingdom, observed that all of the variants that have been shown in this Blog are by and large based on a 9 x 9 Sudoku Grid. This of course is not an accident, since the subtext of the blog is to demonstrate the vast utility of Power (squared) Sudoku White Boards and clipboards, even if (or when) you tire of the original Sudoku puzzle and seek to recapture that old excitement via variations.

Maff has a site called Sudoku Evolution which has created and displays Sudoku variants where the board size is a variable. The site offers a monthly magazine, eBooks, including a free sample pdf file, in which you can modify the file to save or reset your solution entries, as well as print the original puzzle.

The rules for each variant are depicted as graphical animations in an easy and clear manner. Despite there being a few broken links in some of the larger sized 2-D and 3-D puzzles, I plan to spend some time on these kinds of variants. Thanks for letting me know about this site, Maff!

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Crop Circles Sudoku

Crop Circles Sudoku

So as I was waiting for my Toyota Matrix (dashboard view) to be serviced, I was reading fairly metaphysical Book called 2012, The Return of Quetzalcoatl, by Daniel Pinchbeck. who spends some time discussing people who take crop circles seriously. Note the cover art.

It occurred to me that because there have been so many instances of crop circle manifestations, especially in the United Kingdom, that the sheer precision and incuse designs would provide clear examples of geometric one-to-one correspondences with the first 9 digits. Clearly, if aliens are trying to communicate with humanity, using geometric art as the communication medium offers a common reality that beats the symbols we use for our number system(s).

I am indebted to Lucy Pringle’s Crop Circle Photographic Library for these images. Her collection of photographs of these mysterious, aerial views of crops bent artistically is extraordinary.

She is a leading researcher and author of 3 books on the subject — Crop Circles: Greatest Mystery of Modern Times (2000), Crop Circles (2004), and her latest Crop Circles: Art in the Landscape (2007).

I hope this cartoon helps to publicize her efforts to bring serious attention to all humanity (who’s interested) in this curiously ignored, yet significant messaging system between an unknown entity or entities and humanity.

I wish a happy new year to all, worldwide. Whenever you’re ready.

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Akari (Light Up) On Sudoku

Akari (Light Up) On Sudoku

In the Western World, December is often host to festivals of Light. I’ve adapted a puzzle called Akari or Light Up which was originally invented by Nikoli in 2001. His site provides a tutorial for how the rules of the game interplay. Sample puzzles may be found on Nikoli’s Website, which vary in size from 10 x 10 to 36 x 20.

The puzzle shown in the cartoon is fairly easy to solve. It’s a little reminiscent of the old computer game called Minesweeper. Minesweeper offered adjacency clues which included the boxes diagonal to as well as number of exposed sides. When you incorrectly clear a space that was a mine, a rather startling bang! terminates your game. No sound effects for Akari, however.

The Wikipedia article for Minesweeper has a fascinating discussion about patterns and solving strategies by analyses of single and multiple boxes and mine probabilities. Board difficulty measurements are also detailed.

Some online sources of puzzles include: (online player, pdf puzzle generator), 100 Free Akari Downloadable Puzzles at (zipped pdf files), and for online games of varying difficulty.

Have a most enjoyable holiday! I’m glad to enlighten you or at least lighten your mood.

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Do-Sum-Oh (Squiggly) Sudoku

Do-Sum-Oh (Squiggy) Sudoku

I came upon Squiggly Sudoku puzzles at Bob Harris’ site. This variant emphasizes the distortion of the 3×3 squares into irregular geometric shapes also containing 9 squares each. His site contains a proof that n-1 starting numbers (or letters) Du-Sum-Oh (Squiggly) exists uniquely for any n x n sized puzzle. The puzzle above has n starting letters for a 9 x 9 sized puzzle, so it should be easier.

Bob Harris offers various sized puzzles on his website and has published a book called Squiggly Sudoku (Sudoku With A Twist) containing 120 various sized puzzles.

He also provides a useful tutorial about how to solve these puzzles. He cites the Big (and Little) Law of Leftovers. The Big Law of Leftovers: Wherever a group of regions overlaps some rows or columns, the parts outside the overlap (the leftovers) have to be the same.

Another pair of authors, Gideon Greenspan and Rachel Lee provide another book of 200 puzzles, also called Squiggly Sudoku However, of these 200, only 144 are Squiggly Sudoku puzzles, the rest are either classic Sudoku (40) or Samarai Sudoku (16) of various levels of difficulty. They also maintain a Website called Web Sudoku with daily (printable) squiggle and other puzzles.

Another Web Site that provide Squiggly Sudokus is Daily Sudoku By Sam Griffiths-Jones. He is also responsible for several sudoku books, including one for kids and an advanced puzzle book. He also sells Electronic Books of puzzles including Squiggly Sudoku puzzles.

This cartoon is comparable to the cartoon I published last June 2, 2007, called Jigsaw Sudoku where numbers instead of letters were used.

While Sudoku puzzle solving is a great boon to maintaining/increasing cognitive brain function, some may eventually get bored with it, hence the invention of Sudoku variants. The fact that boredom sets in to a previously stimulating activity is just human nature. Similarly doing the same physical exercise initially builds muscle tissue but after a while it no longer does.

There is an indication (See Carved in Sand: When Attention Fails and Memory Fades in Midlife by Cathryn Jakobson Ramin, for example) that our brains, when exposed to greater stimulation have a greater number of healthy nerve cells and stronger connections between them. The act of doing puzzles for example, requires continuing novelty to provide sustained stimulation (and interest). The practical consequence of this is to keep our brains agile throughout our entire lives and not have it turn to oatmeal before we’re ready.

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Playing Card Sudoku

Playing Card Sudoku

Today I am applying the most popular and historically enduring card games using its playing cards to Sudoku. Card Games have been manual endeavors but have tapered off after the 1990s when PCs became sufficiently prevalent. Only since the advent of PCs has digital computer software been written to simulate card games, notably one player Solitaire based games which were a strong motivator of and user interface training, once the PC was available (in offices and homes!).

They are now supplanting physical card games, perhaps most due to the “(double) click” that properly places a card where it should go. This is infinitely more convenient and immeasurably speeds up the game compared with picking up a card and putting it where it belongs. I’m sure some people may “click” on principle just in case the software knows more than they can see (at 3AM).

My favorite visual humor involving Solitaire for the 80s and 90s is shown here.

One way to merge (really small) playing cards with a Sudoku board is to use velcro on all the cards and in each square. Your Sudoku (partial) solutions will persist much longer.

Some interesting Card Game sites include The House of Cards which has information and rules for many kinds of games involving playing cards as well as downloadable software and online versions of card games.

The Card Games Web Site has card and tile games from the world over as well as many links to other card game and card related sites.

Of course, Hoyle’s Games, Improved And Enlarged is a mainstay for Card game rules and mathematical analysis (of odds) since 1835.

Since card games have been around since before 1000 A.D., Where they are believed to have originated in Central Asia and spread to the Moslems and then to Europe and finally to the North and South America and Australia. Look at A Brief History Of Playing Cards” for many fascinating details about the playing card evolution, especially after 1800 in the United States.

The following site also has historical but not necessarily currently played card games:
Rules To Period Games

Thanks to all who are following my whimsical Cartoons involving Sudoku variants.

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Slitherlink On Sudoku

Slitherlink On Sudoku

A big thanks to Jim Bumgardner ( for motivating this Cartoon about Slitherlink puzzles. I had not played this before, but it is quite absorbing. His site provides many sets (books) of 16 Slitherlink puzzles in PDF format, along with their unique solution. They are organized according to level of difficulty. In every even numbered book set, the number counts in each puzzle are symmetrically placed, but the solution is definitely always asymmetrical.

In his instructions, he suggests that dot connections that cannot logically occur should be indicated by a thin x so that the remaining possibilities stand out. In the puzzle above, there would be an x between the 3 and the 0, enabling the 3 to be fenced as shown.

Slitherlink is also known as Takegaki along with other names in Wikipedia. The site also provides solution strategies. Another site that has interesting tutorial is the Nikoli puzzle site. Slow motion animated practice always trumps verbal description. has an FAQ that is very helpful in visualizing what can and cannot be drawn. Puzzles there range from sizes 5×5 to 25×30 in difficulty levels Normal and Hard.

Hirofumi Fujiwara has provided an excellent (non-obvious) “Key to Solution” set of basic, general and strategic rule sets. His site Puzzles And Java World also has puzzles online written in Java including Sudoku, Paint by Numbers, Cross Sums, and Sliding Piece Puzzles.

So don’t sit on the fence! Extend it!

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Arukone (Number Link) Puzzles

Arukone Sudoku

I was alerted to this use of the Sudoku Grid in Yahoo! group’s Sudokuworld about a week ago for a completely different variation exercising mental pathways, which for me, are not well traveled. To make the sample puzzle clearer, I used a yellow path that included both endpoints. There will be 9 such yellow paths, (with none that cross each other,) and at least one blank square long so that all the squares are covered by the (unique) solution. Dark lines may also be used to connect like numbers.

My strategy is (“Get It Surrounded”) to start at the outermost Number (i.e. in a corner) and try to connect to its mate. Then work inwards. If the numbers permit, working from right to left, up to down or the reverse also helps. It is a bit irritating that multiple tentative paths are not easily displayed concurrently.

There are puzzle sites with either paired numbers or paired letters. The grids are anywhere from 5×5 to 16×16. In general, the larger the grid, the more difficult the puzzle is to solve.

Sites with Puzzles and/or Tutorials (in English) include:

Nikoli Number Link Puzzles. The tutorial is terse. Nikoli has 3 books published in Japan containing Number Link puzzles exclusively.

Arukone Puzzles. This site is created and maintained by Vegard Hanssen. It has puzzles in 5×5 and 9×9 and is graded for difficulty. The puzzles are printable and solutions can be displayed (but not printed).

Angela And Otto Janko’s German Puzzle Site, containing many kinds of riddles and puzzles, including Arukone puzzles. The tutorial and strategy section is very concise.

A Wikipedia Article about Number Link Puzzles is likewise quite brief.

Finally, Tim Halbert’s Number Link Site contains an archive of Number Link puzzles (which was generated daily until May 2006). According to Halbert, not all puzzles in the archive have a unique solution. This site also uses the strategy based on working from outside in, to avoid crossed paths. He indicates that Excel can be used as a solving grid, once the numbers are entered. Printing a puzzle out on paper (even multiple times) to solve portably is also a good option.

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Roman Sudoku

Roman Sudoku

There is a continuing fascination with predecessor number systems like Roman Numerals. This Wikipedia site also discusses fractional notation and the subsequent use of the “zero” as N (standing for Null). There are sites which provide Arabic from/to Roman Numeral Converters and those that actually teach how to perform arithmetic operations using Roman numbers.

A little known function in Excel 2004 (maybe others) called roman(arabic_no,form) permits varying Roman Number succinctness. As the help facility notes: we have
0 for classic; 1, 2, 3 for more, more and more concise; and 4 for simplified. (Look up Excel Help for Roman form). In particular the number 499 has 5 (!) versions: CDXCIX, LDVLIV, XDIX, VDIV, and ID.

One of my favorite problems for Computer Programmers learning the C programming language (I taught this in the 1980s) was to write an Arabic to Roman Number Converter, using associative arrays of all the possible symbols as a preferred solution technique. It exercised many language aspects and involved arrays and pointers.

In this Sudoku variant, there is also no room for “small candidate” roman numbers, unless you use a classroom sized whiteboard. Reading clocks with Roman Numerals discloses that IV = IIII and that V, VI, VII and VIII are read while (they or you are) upside down. IV = IIII is done so that IV won’t be confused with an upside down VI, or so they say.

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Barcode Sudoku

Barcode Sudoku

The ubiquity of Barcodes! They’ve come upon the scene suddenly in 1973 and for a while barcodes showed themselves in many inappropriate places (tattoos, cartoons, human foreheads, etc.), they became invisible because they were everywhere. Indeed, they are significantly missing from the educational curriculum. Mathematics ignores them. Ironically, the initial outcry was that human beings should not be treated merely as numbers!

New tagging inventions, particularly the Radio Frequency Identification (RFID) tagging technology were born invisible. The furor against this has been minimal, despite that an RFID tag can be applied to or incorporated into a product, animal, or person for the purpose of identification using radiowaves. Perhaps the mantra for humans could be: I am not a signal tag!

But I digress. I have taught Unix Shell Programming with a Final exam question that was 3 pages of Barcode introductory description and one paragraph of question, involving computing the check digit on bonafide and illegal barcodes.

In the case of this Sudoku variant, it’s clear that the usual solving methods won’t work as well since the graphic representation of the number takes up nearly the entire cell. Perhaps a parallel Sudoku must be solved with ordinary numbers and then the barcode transcription be made.

In any event, I thought this was a cool way to solve a Sudoku puzzle. I hope you enjoy the sight or the struggle.

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Unary Sudoku T

Unary Sudoku T

This is a Unary Number System variation that is popular in China, Japan and Korea. The full Character (Cheng) means honesty or correct according to the Wikipedia Tally marks article.

Something I neglected to mention in the first Unary Sudoku Cartoon is the Encyclopedia of Integer Sequences which contains within it the table of the first 1000 Unary Numbers! This table was created by David Wasserman. While this is something of a non-computational table, it turns out to be really handy when you need to copy and paste the Unary number elsewhere, rather than count the strokes yourself.