Erasable Games Weblog

(Sudoku in words and pictures)

Entries tagged with 'Sudoku Variants'

Hidato™ On Sudoku

Monday, August 25th, 2008

Hidato On Sudoku

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.

Hidato™ appears to be a generalization of Numbrix™ in two senses: Hidato™ distributes the starting numbers over the entire planar grid rather than at the edges and Hidato™ permits paths to be diagonal as well as horizontal and vertical. Both variants have the same objective, to fill in a path that sequentially leads from 1 to the ending number, thereby covering the entire grid without a sequence disconnection or duplication of numbers.

Having the horizontal, vertical and diagonal path choices for the next or previous number reminds me of the (computer) game called Minesweeper. In that game, you are faced with choosing the next cell that doesn’t have a mine hidden in it. In Hidato™, the choice is the next in the sequence being placed properly.

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.

The main site that publishes Hidato™ puzzles is, operated by Gameblend Studios, LLC. This site licenses daily Hidato™ puzzles of varying difficulty to a variety of web based sites, all marred by the same jarring soundtrack, which can eventually be turned off in the options.

I like the Seattle Times version of the Hidato™ site. Its Novice setting shows the next or previous missing number in the sequence on both the grid as well as a linear bar at the bottom of the grid, which is used to highlight any missing number for possible placements. It also offers a checking facility for a partial solution, to keep you on the right track.

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.

See also an article dated July 31, 2008 by Ben Steelman for the about how Hidato™ evolved.

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

Numbrix™ On Sudoku

Monday, July 28th, 2008

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.

Square Root Sudoku

Sunday, July 20th, 2008
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.

Even Sum Sudoku

Sunday, June 29th, 2008

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.

Hellenic Sudoku

Monday, June 9th, 2008

Hellenic Sudoku

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:

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

9 Letter Sudoku

Sunday, June 1st, 2008

9 Letter Sudoku

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!

Superscript Sudoku

Sunday, May 25th, 2008

Superscript Sudoku

Many Web-based Sudoku puzzle sites offer hints in the form of small sized superscripts (like exponents in Algebra). See my favorite site: Seattle Times Sudoku Page. These superscripts are also known as candidate numbers, which represent the remaining possibilities for values in a cell given the starting numbers (which are normal sized and ordinarily displayed).

It occurred to me that it would be very interesting to throw away the starting numbers and just deal with the superscript hints. Some people create these superscripts in all blank cells as a way to solve the puzzle anyway.

What I did is to initially clarify the proper values from the superscript cell entries to the best that I can determine. Then by looking at the blank cells, I noted that superscript values and totally blank cells contained mutually exclusive values. From that I could deduce the value that must be in a blank cell, based on those superscripts along its row, column and block.

It’s an unusual handicap that provides an enjoyable solution process. Solve on!

Inverse Sudoku

Monday, May 19th, 2008

Inverse Sudoku

Like most interesting games, when turning them around by redefining what loses is now what wins, a fascinating variant can be played. For example, in Chess, if the object is to now be the first player to lose all pieces except the king, and a capture is forced if it is available to be played, this yeilds a variant, called Losing Chess that even novice players can excel at.

Another game is called Losing Checkers where you play Checkers according to normal rules (forced to capture if the opportunity arises) but the Loser wins.

In this spirit, I thaought about what it might mean for Sudoku puzzle solving. Since there are so many ways you can fail to solve a puzzle, this didn’t seem to challenging :-)

In this Cartoon variant, We start with a Sudoku solved puzzle and proceed to erase numbers until we have an incomplete or starting puzzle (depending on when you stop erasing). Then try to solve the puzzle that is presented.

If you don’t erase enough numbers, the puzzle may be easily solved. If you erase just enough numbers, (in this case 53 of them), you may either have a uniquely solvable puzzle with a single solution or you may have one with more than one solution. If you keep removing more than 53 numbers (in this case), your ultimate solution may have multiple answers. This means that your starting numbers are “ambiguous”.

There are many programmatic aids in solving sudoku puzzles on the internet, both web based and downloadable applications. One category are Sudoku Puzzle Generators where they produce the starting numbers and you solve. There are also Sudoku Solver Sites, many of which have user interfaces so that *their* puzzle may be solved by you online.

The trick is to find Sudoku Solvers where you get to input the starting numbers. Some sites I found offer:

(1) Excel based Sudoku all in one Generator and Solver by Hari Kumar

(2) Excel based Sudoku-xls Generator, Workpad and Solver by Peter Mladek

(3) AT&T Research Command line: A 9 x 9 Sudoku Solver And Generator And Starting Number analysis

(4) GNUDoQ 0.94 Sudoku Solver

(5) David Ireland’s Number Sudoku Solver (Excel Based)

Happy Erasing!

Kanji Sudoku

Monday, May 12th, 2008

Kanji Sudoku

I was watching the 1st Season, 1st Episode of the U.S. Television Series Heroes on DVD when I noticed a clock with Kanji Characters in a scene with Masi Oka (playing Hiro Nakamura). This led me to do this cartoon.

With non-english alphabets that do not share characters, I find it relatively easy to stare at each glyph and relate it to a sound or number and remember it. (My foreign language skills have otherwise languished.) The games Mah-Jong and Shogi have helped enormously to give me practice in this kind of character recognition.

There are some web sites that facilitate this practice:

I hope solving the puzzle in Kanji doesn’t slow you down too much.

Reversi Sudoku

Sunday, May 4th, 2008

Reversi Sudoku

Earlier today, at my grandson Keegan’s 1st Birthday Party, I talked with a friend of mine (and another Keegan Grandparent) who sells books. Old books. He calls it: Lou Manrique — Antiquarian Bookseller. He talked about his recent passion for playing Reversi online with folks worldwide (but one at a time). Wikipedia goes into great detail about Reversi.

A long time back, I played the same game under the name Othello, something the Japanese trademarked back in the early 1970s. But in practice, Reversi and Othello are synonymous. Interestingly, There is a World Othello Federation.

Lou got me to thinking about how I could combine Reversi and Sudoku. Out of that comes this cartoon today. I imagine that you can solve the sudoku puzzle, but as you add a number to the board, you now also have to choose a black or white number. As you complete the puzzle, the board gets more fully filled in, perhaps by alternating black or white pieces.

Once the puzzle is solved, Reversi rules take effect.

A legal move in reversi is one which will complete linear envelopment at least one opponent’s disc and convert the disk(s) to your initial color. Play passes to your opponent if you have no legal moves. No legal moves for both players means the game ends and your territory extent is counted.

Now starting from the interior, it is necessary to flip linearly enveloped opposite colored numbers to your initial color (black). You alternate flipping to black, then to white, until there are no more flips possible. Then notice which color is more popular. It’s probably much better with two people playing.

To (re)freshen your skills playing Reversi, there is a computer Applet which you can play: See Reversi Java Applet By Thomas Wolf.

Consider solving this Sudoku Puzzle variant as bait for getting someone to help you finish the Reversi portion.

There is also a Reversi variation you can play in this! It’s called Reversed Reversi, where the goal is to minimize territory (i.e. lose). Just let your opponent know your objective, so that the game is more interesting.