Erasable Games Weblog

(Sudoku in words and pictures)

Superscript Sudoku

May 25th, 2008 by arkay

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

May 19th, 2008 by arkay

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

May 12th, 2008 by arkay

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

May 4th, 2008 by arkay

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.

Domino Pairs Sudoku

April 28th, 2008 by arkay

Domino Pairs Sudoku

The last cartoon that referred to Dominos and Sudoku occurred on March 30, 2007. At the time, I was more concerned with the number representation shown by the spots or “pips” than the tiles or “bones”.

So with this cartoon, I offer a variant using the tiles of a double nine set of Dominos.

A little analysis shows that a 9 by 9 Sudoku board permits a filled in arrangement of 40 tiles, keeping 1 cell uncovered. In this cartoon, I choose that cell to be the center cell, identified and covered by a circular single pip. For clarity, the starting tiles that are horizontal, have their border tinged in bright blue. Vertical tiles are shown in black.

Unfortunately, a double nine set of dominos contains 55 tiled pieces, of which 19 are inappropriate for Sudoku uniqueness rules (all tiles containing a blank and all the symmetric pairs). That leaves up to 36 unique tiles to place. This means there will be at least 4 tiles that are “cloned” in order to solve the Sudoku puzzle.

Fortunately, the tiles with pips are rotationally symmetric, unlike numbers printed on tiles, so they can fit into whateveer sudoku nook is required.

In keeping with the “solving rules” of this variant, place tiles or adjacent numbers two at a time. This is an unusual way of solving sudoku, especially at the beginning when you may be hard-pressed to find even a single number.

Perhaps you should invest in a double nine domino set as a solving aid and have a paper copy of the puzzle in progress to record your emerging solution, and place tile pairs only as they become available.

Sudoku XV

April 20th, 2008 by arkay

Sudoku XV

The 3rd World Sudoku Championship (with classic and variant puzzles) has been won by Thomas Snyder of the United States. In second place is Yuhei Kusui of Japan and Jakub Ondrousek of the Czech Republic who placed third. Mr. Snyder also won the Classic Sudoku Championship (no variants), followed by David McNeill of the United Kingdom in second place and Michael Ley of Germany in third place. Team competition results show the Czech Republic in first place, followed by Japan and Germany in 2nd and 3rd place, respectively.

One of the variants shown in the Instruction booklet (First Link in the Downloads Section, a PDF file) for this year’s contenders is known as Sudoku XV. The fewer numerical clues are offset by number pair relationships as sums of 5 or sums of 10. You may use the full force of sudoku analysis to fill in numbers and get the bonus of a neighboring pair value.

There seems to be few puzzle sources for this variant except in Competition instruction booklets. I hope this is temporary. It’s an interesting blend between a limited Kakuro and Sudoku.

Extra Regions Sudoku

April 13th, 2008 by arkay

Extra Regions Sudoku

The idea for this Sudoku variant comes from the instruction booklet for the national Finals offered on March 23, 2008. While their example in the booklet doesn’t give any starting numbers, I thought that toorestrictive, so I do offer them. This is found on the World Sudoku Championship 2008.

In this age of the internet, everyone is (unnaturally) interested in everyone else’s opinion. You get it via Email (spam), Pop-up Windows, home page portals. So for those for whom answering surveys is an avocation, I offer the 1 question, universal question survey. To add to your convenience, the question is offered for your personal reflection, so you needn’t send it anywhere (especially here).

Last month, I noticed on the Usenet Newsgroup rec.puzzles, a stirring discussion about the question by Anthony Buckland, of:

Is the popularity of Sudoku dependent on computers?

Not for solution, but for creation. Can a new Sudoku for the newspaper each day reasonably be produced with nothing more than pen-and-paper technology? By, in the same way as crosswords and other puzzles, one person in the time that a newspaper (chain, if you like) fee would make worthwhile?

This may conform to a generational answer. Those who lived all their lives with computers may not be able to imagine life without them and underestimate human ingenuity. Those like me who fully embraced Computers starting in 1962, as an adult, know full well, computers are unnecessary for this endeavor. Many others, may consider this question a definite maybe.

As a simple example, it might take time to create a solved sudoku puzzle. But once done, a bit more time to create a set of starting numbers leading to one solution. Once you have this, you can create, using typesetting machinery, many sudoku puzzles from that one solution: Rotate the grid 90 degrees, Permute each component of vertical or horizontal blocks (e.g. Row 1 –> Row 3; Row 2 –> Row 1; Row 3 –> Row 2). Numbers can be mapped to other numbers for different puzzles.

In fact, of the 6,670,903,752,021,072,936,960 possible unique results that offer exactly one solution. The same puzzle looks totally different, when successively rotated 90 degrees, or by mirroring (reflection) or by relabeling the digits. When these effects are ignored, there are merely 5,472,730,538 essentially different Sudoku grids.

Questions That arise:

Can you trust a Sudoku that is created by hand? (Misprints happen even now)

How many puzzles can be generated per day, per week etc, if done only by hand?

Would there be as much (or more) starting number symmetry with hand puzzles?

Check out the 2008 World Sudoku Championship starting this week.

Cross Sums Sudoku

April 7th, 2008 by arkay

Cross Sums Sudoku

So the question is: Does giving totals for all outer 3 cells (across and/or down) in selected 9×9 squares give the ability to solve a sudoku puzzle with a reduced collection of starting numbers? Because the interior (to the left of the 4th column and above the 4th row) has no space to specify their sums, I’ve chosen normal starting numbers instead.

The book by Xin-She Yang, Ph.D., called Cryptic Kakuro and Cross Sums Sudoku offers cross sums of all the 9×9 squares by creating wider detachments between them. There are no starting numbers as a result.

This puzzle also differs from traditional Kakuro Puzzles in that the entire 9×9 grid is used. In Kakuro, only a subset of the grid is used. See my cartoon on Kakuro last May 27, 2007.

Although I tried to offer all possible sums (combinations) of 3 unique single digits 1 through 9, there are too many to list. Trust in symmetry to help furnish the rest of them, if needed.

In other news, the 3rd Annual World Championship Sudoku Competition has just provided some downloadable materials for study and sample variations that (we and) the contestants may work on. Note that in the individual competition, Sudoku variants score double the points of Classical Sudoku puzzles where the expected solving time is the same.

Mirror Sudoku

March 30th, 2008 by arkay

Mirror Sudoku

There seem to be three kinds of mirroring: What you see in a mirror, mimicry and instant object replication. The first depends on symmetry, the second depends on synchronicity and the third on identity. Andrew Carson of Dungeon7Sciences discusses what a mirror image is about and why the mirror doesn’t show us upside down as well.

In psychological or theatrical terms, mirroring (mimicry) is concurrently copying what someone else is doing (perhaps while communicating with them). Group synchronized dancing (or swimming) are good examples of this. It is called satire when the mimicry is delayed.

In Computer terms, a mirror is an exact copy of data of some extent of storage. On the Web, mirrors contain repositories of desired software or replicated data, useful when there is a surge in demand for a particular article, picture, software program or datum.

There is a description of Mirror Sudoku which requires that the Sudoku puzzle starting number placements be symmetrical horizontally and/or vertically. This puzzle with its starting number positions conforms to horizontal and vertical symmetries.

Remember to draw the numbers slowly and precisely. It’s an alternate universe you’re trying for.

Fillomino On Sudoku

March 23rd, 2008 by arkay

Fillomino On Sudoku

Nikoli invented this puzzle in 2001. Once called Allied Occupation, Fillomino is a merger of the words Filled and Polyomino. A Polyomino describes the shapes assumed of a varying number of multiple cells, that have at least one side touching another cell.

Wikipedia describes this as a logic puzzle and gives more elaborate rules, particularly about the possibility that 2 or more given numbers could belong to the same pentomino. Solution methods are also offered.

As this is a fairly popular puzzle in its own right, there are a variety of Websites that offer Fillomino:

  • Puzzle Club a British Site offers Many Math, Number, Logic, and Word Puzzles in addition to Fillomino. Registration is 10 Pounds/Year
  • Vegard Hanssen’s Fillomino Puzzles, which has more than 16,000 Puzzles in its database. Mr. Hanssen is a perl programming language afficionado, See his perl scripts for generating his name(s).
  • provides 8x8s, 12×12 and 16×16 size grids for their daily Fillomino puzzles.
  • Simon Tatham has a similarly named puzzle called Filling on his website.
  • Finally, has an interesting discussion about Fillomino Skyscrapers which is a mashup of Fillomino and what I called buildings. The numbers on the outside are clues to how many “buildings” can be viewed from that location. Higher valued numbers block lower valued numbers, but offer interesting clues about the composition of the row or column. See my Sudoku Buildings Cartoon posted last July 15, 2007.

Get your fill of Fillomino! (but take your time.)