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Numbrix™ On Sudoku

Numbrix(tm) On Sudoku

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 NumbrixTM.

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.

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Square Root Sudoku

Square Root Sudoku: Disguise The Units Digits, Complete the Puzzle And Be Radical!

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: 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.

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Even Sum Sudoku

Even Sum Sudoku

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.

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Wormy Sudoku

Wormy Sudoku

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 ( containing a huge number of variant Sudoku puzzles: Sudoku X Worms and which heralds the upcoming Mathfest (August) 2008

No doubt, the above puzzle is fairly easy to solve compared to those on 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!

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Unary Sudoku

Unary Sudoku

The most rudimentary (prior to computers) number system is the Unary number system, consisting purely of 1s or strokes, usually representing tallies. It is described as a base 1 number system.

I studied it in my Computer Science classes (way back when) in a course about Computability, Turing Machines and Automata Theory. In particular, the Turing Machine example described a machine readable tape containing groups of 1s each separated by a single 0.

I used it as a child/teenager to count up physical items in several categories (popularity counts) and for keeping score in pool. Remember that this was *before* computers existed. Also, not all pool halls had an overhead Abacus. I was much more intimate with the physical world then.

The problem, as I see it, in this variant of Sudoku is that there’s absolutely no space to insert candidate (little) numbers based on which logic is performed. One possibly impractical solution is to do this on a classroom size whiteboard. But I warn you: Don’t have a stroke!

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Binary Sudoku 10

Binary Sudoku 10

Underneath it all, binary numbers are the numerical lingua franca of all computers! In mathematics, most rational numbers can be represented as binary numbers, provided that their binary places (bits) do not exceed the computer word length (or some multiple of it).

For this puzzle, I’ve translated the first 9 integers to their equivalent values written as binary numbers. Since most human beings lock onto the first number system they ever learned (decimal or base 10 numbers), they entertain other number systems only with painful difficulty, unless paid (as computer programmers/developers) not to do so.

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Odd Sudoku

Odd Sudoku

I had to skip a week due to superceding short term tasks to complete involving a Linux workshop I conducted last week.

This Sudoku variant supplies the correct odd/even location of the numbers of the solved puzzle, but offers fewer starting numbers. This encourages constraining logic when considering the possibilities of each cell.

A discussion of Odd and Even Numbers is offered in Wikipedia.

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Linear Sudoku

Linear Sudoku: Tickertape Or Mobius Strip?

The idea for using a string of 81 single digits [the solution] or 81 characters including the blank character [the unsolved puzzle] probably originated on Usenet due to the text based messaging used there. I’ve always been fascinated with Mobius Strips and Klein Bottles (the 3-D analogue). A very enjoyable, math oriented book first published in 1958, Edited by Clifton Fadiman, called Fantasia Mathematica conjectures about the possibilities raised by these strange objects, among other science fiction topics.

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MathTrek And Brainfreeze

I’ve been subscribing to Science News since 1995. [As I get older, Science (and Religion) become more fascinating.] It’s an excellent weekly news magazine of science, usually about 16 pages with minimal ads and concise (1-2 page), up to date articles. One of the features of Science News is an occasional column (blog) written by Ivars Peterson, called Math Trek Blog.

Mr. Peterson has written several articles about Sudoku for the magazine, notably:
Sudoku Math (June 2005),
First World Sudoku Championsip (March 10-11, 2006),
Pentomino Sudoku (May 2006) and
Sudoku Class (February 2007).

His articles focus on aspects of Sudoku in a clear, concise way and unlike other (opinion) columns, provide a bibliography to other sources of information. His latest, entitled Sudoku Class, provides information about Sudoku in classrooms, Math conferences and describes a Website by Dr. Laura Taalman (of James Madison University) called Brainfreeze Puzzles.

Although not updated in 10 months, Dr. Taalman’s site describes eleven Sudoku variants with examples. She is co-authoring (with Philip Riley) a new book called: Color Sudoku to be published in May 2007, containing examples of these variants. (Perhaps that is the reason for the frozenness [no pun intended] of her website.)

On her University website, Dr. Taalman conducted and preserved a Sudoku Puzzle Problem of the Week contest from January through April 2006. Thirteen downloadable Sudoku variant puzzles (with solutions also available) are shown there. I haven’t tried them yet but plan to.