There’s been a significant increase in traffic to this blog ever since I posted the Numbrix puzzle variant at the end of July. For that, I thank you all for visiting.

This week, Barnes and Noble displayed a book by Dr. Gyora Benedek, called Hidato (A Hebrew word meaning: My Puzzle). Dr. Benedek writes an excellent (short) autobiography. The book itself contains variously sized and shaped grids including non-square (and non-quadrilateral) ones. Some web-based 10 x 10 grids have an interesting characteristic in that the middle 4 cells are darkened, indicating that they are not in play, leaving 96 cells to the playing area.

Typically, you’ll be connecting sequence segments based on the starting numbers. Luckily, the connectedness of the overall sequence requires that local neighborhoods of numbers be near in value. What provides a challenge is that It is possible that multiple such sequences may be intermixed.

On the negative side, there’s no facility for saving the current state of the puzzle (without printing it) and when you print the puzzle out, it does not let you return to the puzzle to continue with it.

Have a good time with this variant. This is just the beginning for it.

]]>A while back, I created a Cartoon called Double Sudoku. In that variant, there were 2 totally separate Sudoku Puzzles embedded in a single Sudoku Grid with all split cells.

The current cartoon is reminiscent of that one, but has N-1 starting numbers that are the same for both puzzles. (Double Sudoku had all different starting numbers for each puzzle.)

In researching where this variant may exist on the internet, I came across a site called SuBundle.com, by Dr. Chen, which has generalized this concept even further. Provided are puzzles with as many as 256 possible puzzles generated from multiple “Key Cells” in a single grid.

Normally, a “Key Cell” either contains a single digit M, which signifies that there are that (M) many puzzles bundled together (and you must figure out which digits lead to a solution) or the “Key Cell” contains multiple digits, which means that each of those digits leads to a different Sudoku puzzle.

Truly, this is a **“Green”** website for Sudoku Puzzles, where the number of Sudoku Grids is severely conserved. His site even offers multiple blank Sudoku grids that may be printed from a pdf or png formatted file.

I’ve provided tips for solving the above puzzle in the single grid by splitting blank cells in two parts. Of course, you can write each puzzle out on paper or duplicate electronically and solve each normally (and one after another).

I hope you enjoy solving this example of Sudoku puzzle cousins twice removed!

]]>This cartoon has at its root, a cartoon from 3/4/2007, called Linear Sudoku. For those that wish to describe a Sudoku puzzle to others, particularly by Email, the best way is to create either an 81 character text string or a 9 strings of size 9, where the dot represents the blank cell in both cases.

The site: Sudoku @ Paulspages permits the selection of a Sudoku puzzle, be it random, non-symmetric or from a gallery of puzzles and from that point, it is possible to export it as a text file as shown to the right of the Board in this Cartoon.

There are several benefits for this format, not the least of which is communicating with another puzzler to verify a difficulty or a solution. For those who are visually impaired, they can employ a software program that converts text to speech, thereby permitting the person to hear what the puzzle elements are, so that they can be solved mentally, or transcribed to a braille-writer for reference.

Now that text messaging is becoming most popular, these text strings can convey a puzzle in a minimum of words. This can be useful in timed competitions, where the winning solution is “texted” over to the contest judges for least time and solution correctness. See, for example:

Sudoku Text Challenge sponsored by the TimesOnline and retrieved June 17, 2008.

On July 13, 2008, Marilyn Vos Savant, in her column Ask Marilyn, (which is syndicated through Parade Magazine) introduced a number puzzle which helps focus the mind. It’s called Numbrix^{TM}.

According to Michael Keller, of Solitaire Laboratory, who commented in the newsgroup rec.puzzles on July 22, 2008:

This puzzle dates back at least to 1981, when Steve Wilson (who I think invented it) published some puzzles in Games Magazine (July/August 1981, page 38). Steve’s puzzles are much harder and more interesting than Marilyn’s pathetic examples.

The puzzle in its present incarnation seems easy to those with much numerical puzzle experience, but can be challenging to those people with number anxiety and/or with interrupt-driven lives.

Although Marilyn’s puzzles are on 7×7 and 8×8 grids, there’s no reason why a 9×9 grid can’t be fully used. I restrained myself from making it harder by eliminating selected edge numbers, but that could be done by you, when copying the initial puzzle.

Stay on the path and be enlightened (or at least delighted), once you finish the puzzle.

]]>I received a wonderful (analog) watch from my niece. It was a Square Root Watch! As a math instructor this made me smile and be pleased with my niece for thinking of me.

The watch is really a low-tech encryption of time, especially for those whose instant reaction to any math-related symbol is anxiety. Look at: signals.com if you’d like to acquire a watch like this and also impress your friends. It’s has made everyone laugh who I have showed it to. Of course, the watch wearer may have been the source of amusement. hah!

As a result, I adapted this concept to Sudoku puzzling. The result is the cartoon for today. Enjoy the puzzle but don’t be absurd.

]]>My wife pointed out that in no way are solving the puzzle and making a tune in a cause and effect relationship. In the interest of clarity, what I really mean is for you to solve the puzzle and then sing, whistle, hum, play, and/or listen to any tune of your choice, composed by you or anyone else. This way, all can take part.

As I look at these musical symbols, I am aware that these look perfectly natural to me, since I learned to read piano music at the age of 5. I’m sure there are those for whom these symbols are mostly strange.

Music notation and symbols have long term strategic importance for preserving musical compositions for future generations. It is a true encoding. Various media containing musical sounds: Cylinders of the early 20th century, 78 RPM phonograph records of the 1920s-1950s, 45 RPM records (1950s), 33.3 RPM long playing records (1950-1980s), Audio Tape (1960-1980s), and CDs (1980-now), MP3s (2000-now) may appear to be lasting, but just remember that all of them require a device to play the media. If the device is not available, the physical media become merely fashion statements and the electronic media become indecipherable. Musical notation on sheet music persists!

Enjoy your summer, now that it is finally here! Muse on the puzzle.

]]>Today’s puzzle variant comes from the 2008 Sudoku World Competition Instruction booklet. I’ve renamed it Even Sum Sudoku for clarity. About a year ago, I published a cartoon called Odd Sudoku, where either Odd or Even contiguous Cells of at least size 2 were offered. This is not like that.

I’ve eliminated some starting numbers from the original puzzle and identified the cells in yellow as pairs with values summing to an even result.

One question that occurs to me is: what is the probability of having Even Sum Pairs for all the arrangements of this puzzle? Obviously, there are at least 8 Even Sum Pairs that have already been earmarked. From previous calculations (Domino Sudoku Cartoon), excluding the center cell, there are 40 pairs of contiguous cells in an arrangement (and there are 2**4 = 16 pair arrangements since:

- In Row 1, columns 1 and 2 [A = Across] or the other starting in Column 1, Rows 1. 2 [D = Down]
- In Row 2, columns 2, 3 [A] or Column 2, Rows 2, 3 [D]
- In Row 3, columns 3, 4 [A] or Column 3, Rows 3, 4 [D]
- In Row 4, columns 4, 5 [A] or Column 4, Rows 4, 5 [D]

Since each of these can be selected independently, there are 2*2*2*2 = 16 arrangements.

For any arrangement, how many are Even Pairs are there? It turns out, once you’ve solved the puzzle, you can count:

Across [A] | Down [D] | |

Odd: 8 Even: 8 | Odd: 10 Even: 6 | |

Odd: 9 Even: 3 | Odd: 7 Even: 5 | |

Odd: 3 Even: 5 | Odd: 5 Even: 3 | |

Odd: 3 Even: 1 | Odd: 3 Even: 1 |

Totals: | Odd: 23 Even: 17 | Odd: 25 Even: 15 |

Arrangement | No. | Odd | Even |

AAAA | 1 | 23 | 17 |

DDDD | 2 | 25 | 15 |

ADAA | 3 | 21 | 19 |

AADA | 4 | 25 | 15 |

AAAD | 5 | 23 | 17 |

DAAA | 6 | 25 | 15 |

ADDA | 7 | 23 | 17 |

ADAD | 8 | 21 | 19 |

AADD | 9 | 25 | 15 |

DDAA | 10 | 23 | 17 |

DADA | 11 | 27 | 13 |

DAAD | 12 | 25 | 15 |

ADDD | 13 | 23 | 17 |

DADD | 14 | 27 | 13 |

DDAD | 15 | 23 | 17 |

DDDA | 16 | 25 | 15 |

Frequency | Odd | Even | |

2 | 21 | 19 | |

6 | 23 | 17 | |

6 | 25 | 15 | |

2 | 27 | 13 |

P(Even = 19) = .125

P(Even = 17) = .375

P(Even = 15) = .375

P(Even = 13) = .125

A nice discrete, symmetric binomial distribution! Enjoy getting even.

]]>After the rain or flooding, there will be worms that surface to find a new nesting place.

Where I live, rain has been a constant companion since last October. Other parts of the world (Myanmar, China, near the Mississippi River) have significantly more extreme weather and climate. My hat is off to them abiding under such conditions. I am sad for those who have lost loved ones, pets, possessions and stability they worked so hard for to flooding.

Laura Taalman’s Brainfreeze Puzzle Math Variants contains a variation called Worm Sudoku (in green type on that page). I have varied it further to not offer sequence direction nor offer two cell sequences nor different color (nor turning) worms.

Other sites with a worm variation include a German site (sachsentext.de) containing a huge number of variant Sudoku puzzles: Sudoku X Worms and Threesixty360.wordpress.com which heralds the upcoming Mathfest (August) 2008

No doubt, the above puzzle is fairly easy to solve compared to those on Brainfreezepuzzles.com. My wife has insisted that I offer more easily solved puzzle examples, since the variations are confounding enough. So here it is.

If this puzzle makes you squeamish, consider the real thing: Sudoku Worm!

]]>Because of the natural correspondence of sequential alphabet lists of letters to positive integers, many non-english alphabets have a strong link to number sequences. The Hellenic (Greek) letters offer a numeric code that can be used to vary Sudoku puzzles.

In reviewing the connotations of this alphabet and its relation to Mathematics (and Statistics), I discovered several interesting sites:

- Learning The Greek Alphabet In 10 Minutes (a youtube.com 9 minute film)
- Greek Letters Used in Mathematics
- The obsolete letter Digamma, which is the original 6th letter of the Greek Alphabet

Since I used the Font Face MMa Greek Bold, it automatically translated the number 6 into a right-to-left flipped 3. The original Digamma was more like an F or f. Go figure (so to speak).

This puzzle is considered more than medium difficulty, but if you have become facile in manipulating 9 symbols, the essence of the Sudoku logic should produce a solution sooner than later.

“Wonder is the beginning of wisdom.” — Greek Proverb

]]>In reviewing the accumulating collection of Sudoku variants, I noticed that I never offered the “straight-forward” mapping of letters for the digits 1 through 9. I hereby remedy this oversight with this cartoon.

This puzzle is also the outgrowth of a “commission” for me to produce a Sudoku puzzle with special letters spelling out a company name and project acronym. This puzzle is to be used as something to do while listening to how wonderful the future of specific software will be. As always, any feature of the future indicates what is not a feature of the present software. (Once upon a time, this too was future shock and awe.)

This puzzle is rated hard, especially using letters, if you are not used to them. Good luck!

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