LA HISTORIA

Automorphic sudokus are sudoku puzzles which solve to an automorphic grid. A grid is automorphic if it can be transformed in a way that leads back to the original grid, when that same transformation would not otherwise lead back to the original grid. An example of a grid which is automorphic would be a grid which can be rotated 180 degrees resulting in a new grid where the new cell values are a permutation of the original grid. An example of an automorphic sudoku which solves to such a grid is below[71] .

Mathematics of Sudoku 162

Notice that if this sudoku is rotated 180 degrees, and the clues relabeled with the permutation (123456789) ->

(987654321), it returns to the same sudoku. Expressed another way, this sudoku has the property that every 180 degree rotational pair of clues (a, b) follows the rule (a) + (b) = 10.

Since this sudoku is automorphic, so too its solution grid must be automorphic. Furthermore, every cell which is solved has a symmetrical partner which is solved with the same technique (and the pair would take the form a + b = 10).

In this example the automorphism is easy to identify, but in general automorphism is not always obvious. There are several types of transformations of a sudoku, and therefore automorphism can take several different forms too.

Among the population of all sudoku grids, those that are automorphic are rare. They are considered interesting because of their intrinsic mathematical symmetry.

See also

• Sudoku - main Sudoku article

• List of Sudoku terms and jargon

• Dancing Links - Donald Knuth

• Algorithmics of sudoku

External links

• Enumerating possible Sudoku grids [72] by Felgenhauer and Jarvis, describes the all-solutions calculation

• V. Elser's difference-map algorithm also solves Sudoku [73]

• Sudoku Puzzle - an Exercise in Constraint Programming and Visual Prolog 7 [74] by Carsten Kehler Holst (in Visual Prolog)

• Sudoku Squares and Chromatic Polynomials [75] by Herzberg and Murty, treats Sudoku puzzles as vertex coloring problems in graph theory.

Mathematics of Sudoku 163

References

[1] http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL87-2.pdf [2] http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf [3] http://www.afjarvis.staff.shef.ac.uk/sudoku/

[4] http://www.afjarvis.staff.shef.ac.uk/sudoku/sudgroup.html [5] http://www2.ic-net.or.jp/~takaken/auto/guest/bbs46.html [6] http://www.csse.uwa.edu.au/~gordon/sudokumin.php [7] http://www.sudoku.com/boards/viewtopic.php?p=2

[8] http://www.sudoku.com/boards/viewtopic.php?t=44&start=138 [9] http://www.sudoku.com/boards/viewtopic.php?t=44&start=527 [10] http://www.sudoku.com/boards/viewtopic.php?p=2992 [11] http://www.sudoku.com/boards/viewtopic.php?t=44&start=412 [12] http://www.sudoku.com/boards/viewtopic.php?t=44&start=422 [13] http://www.sudoku.com/boards/viewtopic.php?t=44&start=430 [14] http://www.sudoku.com/boards/viewtopic.php?t=4835&start=3 [15] http://www.sudoku.com/boards/viewtopic.php?t=2511&start=17 [16] http://www.sudoku.com/boards/viewtopic.php?t=44&start=568 [17] http://www.sudoku.com/boards/viewtopic.php?t=44&start=525 [18] http://www.sudoku.com/boards/viewtopic.php?t=44&start=567 [19] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=2 [20] http://www.sudoku.com/boards/viewtopic.php?t=44&start=41 [21] http://www.sudoku.com/boards/viewtopic.php?t=44&start=527 [22] http://www.sudoku.com/boards/viewtopic.php?p=2

[23] http://www.sudoku.com/boards/viewtopic.php?t=2840 [24] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=11 [25] http://www.sudoku.com/boards/viewtopic.php?t=44&start=412 [26] http://www.sudoku.com/boards/viewtopic.php?t=44&start=548 [27] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=1 [28] http://www.sudoku.com/boards/viewtopic.php?t=44&start=525 [29] http://www.sudoku.com/boards/viewtopic.php?t=44&start=543 [30] http://www.sudoku.com/boards/viewtopic.php?t=44&start=543 [31] http://www.sudoku.com/boards/viewtopic.php?t=44&start=543 [32] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=14 [33] http://www.sudoku.com/boards/viewtopic.php?t=44&start=567 [34] http://www.sudoku.com/boards/viewtopic.php?t=44&start=567 [35] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=2 [36] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=2 [37] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=5 [38] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=11 [39] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=11 [40] http://www.sudoku.com/boards/viewtopic.php?t=2840&start=11 [41] http://www.sudoku.com/boards/viewtopic.php?t=44&start=393 [42] http://www.sudoku.com/boards/viewtopic.php?t=44&start=187 [43] http://www.sudoku.com/boards/viewtopic.php?t=44&start=403 [44] http://www.setbb.com/phpbb/viewtopic.php?t=366&mforum=sudoku [45] http://www.sudoku.com/boards/viewtopic.php?p=3366

[46] http://homepages.cwi.nl/~aeb/games/sudoku/nrc.html [47] http://www.sudoku.com/boards/viewtopic.php?t=44&start=454 [48] http://www.afjarvis.staff.shef.ac.uk/sudoku/sud23gp.html [49] http://www.maths.qmul.ac.uk/~pjc/preprints/sudoku.pdf [50] http://www.afjarvis.staff.shef.ac.uk/sudoku/sud24gp.html [51] http://www.sudoku.com/boards/viewtopic.php?t=4281 [52] http://www.sudoku.com/boards/viewtopic.php?t=4281&start=6 [53] http://www.math.ie/checker.html

[54] http://www.csse.uwa.edu.au/~gordon/sudokupat.php?cn=3 [55] http://www.sudoku.com/boards/viewtopic.php?t=3284&start=7 [56] http://magictour.free.fr/sudoku6

[57] http://www.sudoku.com/boards/viewtopic.php?t=2082&postdays=0&postorder=asc&start=16

Mathematics of Sudoku 164

[58] http://www.sudocue.net/minx.php

[59] http://homepages.cwi.nl/~aeb/games/sudoku/nrc.html [60] http://www.bumblebeagle.org/dusumoh/9x9/index.html [61] http://www.bumblebeagle.org/dusumoh/proof/index.html [62] http://www7a.biglobe.ne.jp/~sumnumberplace/79790008/

[63] http://www.sudoku.com/boards/viewtopic.php?p=11444#11444 [64] http://www.afjarvis.staff.shef.ac.uk/sudoku/ed44.html [65] http://www.sudoku.com/boards/viewtopic.php?t=44&start=138

[66] http://www.sudoku.com/boards/viewtopic.php?t=5384&postdays=0&postorder=asc&start=0 (ask for some patterns that they don't have puzzles)

[67] http://www.sudoku.com/boards/viewtopic.php?t=4209&highlight=largest+empty+hole largest hole in a sudoko; largest empty space [68] http://www.flickr.com/photos/npcomplete/2471768905/ large empty space

[69] http://www.sudoku.com/boards/viewtopic.php?t=1180&postdays=0&postorder=asc&start=15 largest number of empty groups?

[70] http://www.flickr.com/photos/npcomplete/2361922691/ clues bunched in clusters [71] http://www.flickr.com/photos/npcomplete/2629305621/''(Six Dots with 5 x 5 Empty Hole) [72] http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf

[73] http://plus.maths.org/latestnews/jan-apr06/sudoku/index.html [74] http://www.visual-prolog.com/vip/example/pdcExample/sudoku.htm [75] http://www.ams.org/notices/200706/tx070600708p.pdf

Kakuro

An easy Kakuro puzzle

Kakuro or Kakkuro (Japanese: カックロ) is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword. Kakuro puzzles are regular features in most, if not all, math-and-logic puzzle publications in the United States. Dell Magazines came up with the original English name Cross Sums and other names such as Cross Addition have also been used, but the Japanese name Kakuro, abbreviation of Japanese kasan kurosu, (加算クロス, addition cross) seems to have gained general acceptance and the puzzles appear to be titled this way now in most publications. The popularity of Kakuro in Japan is immense, second only to Sudoku among Nikoli's famed logic-puzzle offerings.[1]

The canonical Kakuro puzzle is played in a grid of filled and barred cells, "black" and "white"

respectively. Puzzles are usually 16×16 in size,

although these dimensions can vary widely. Apart from the top row and leftmost column which are entirely black, the grid is divided into "entries" — lines of white cells — by the black cells. The black cells contain a diagonal slash from upper-left to lower-right and a number in one or both halves, such that each horizontal entry has a number in the black half-cell to its immediate left and each vertical entry has a number

Kakuro 165

Solution for the above puzzle

in the black half-cell immediately above it. These numbers, borrowing crossword terminology, are commonly called "clues".

The object of the puzzle is to insert a digit from 1 to 9 inclusive into each white cell such that the sum of the numbers in each entry matches the clue associated with it and that no digit is duplicated in any entry. It is that lack of duplication that makes creating Kakuro puzzles with unique solutions possible, and which means solving a Kakuro puzzle involves investigating combinations more, compared to Sudoku in which the focus is on permutations. There is an unwritten rule for making Kakuro puzzles that each clue must have at least two numbers that add up to it. This is because including one number is mathematically trivial when solving Kakuro puzzles; one can simply disregard the number entirely and subtract it from the clue it indicates.

At least one publisher[2] includes the constraint that a given combination of numbers can only be used once in each grid, but still markets the puzzles as plain Kakuro.

Some publishers prefer to print their Kakuro grids exactly like crossword grids, with no labeling in the black cells and instead numbering the entries, providing a separate list of the clues akin to a list of crossword clues. (This eliminates the row and column that are entirely black.) This is purely an issue of image and does not affect solving.

In discussing Kakuro puzzles and tactics, the typical shorthand for referring to an entry is "(clue, in numerals)-in-(number of cells in entry, spelled out)", such as "16-in-two" and "25-in-five". The exception is what would otherwise be called the "45-in-nine" — simply "45" is used, since the "-in-nine" is mathematically implied (nine cells is the longest possible entry, and since it cannot duplicate a digit it must consist of all the digits from 1 to 9 once). Curiously, "3-in-two", "4-in-two", "5-in-two", "43-in-eight", and "44-in-eight" are still frequently called as such, despite the "-in-two" and "-in-eight" being equally implied.