Easy Cross Sums puzzle

Cross Sums is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword. In principle, Cross Sums puzzles are integer programming problems, and can be solved using matrix techniques, although they are typically solved by hand. Cross Sums are regular features in most, if not all, math-and-logic puzzle publications in the United States. Dell Magazines came up with the name Cross Sums (although they now use the name Kakuro), and other publishers also use it now. Other names, such as Cross Addition, are also used. In Japan, the puzzle is called Kakuro, an abbreviation of kasan kurosu (加算クロス, addition cross); now, the name Kakuro has come to be at least as widely used worldwide as the name Cross Sums. The popularity of Kakuro in Japan is immense, second only to Sudoku among Nikoli's famed logic-puzzle offerings. In an international tapdance, Kappa reprints Nikoli Kakuro in the United States, in GAMES Magazine under the name Cross Sums. The Guardian in Britain began printing the puzzle under the name Kakuro in September 2005; since then many other British papers have followed suit and now also print daily puzzles. Image File history File links Crosssum. ... Image File history File links Crosssum. ... A logic puzzle is a puzzle deriving from the mathematics field of deduction. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Transliteration is the practice of transcribing a word or text written in one writing system into another writing system. ... The crossword is the most common variety of word puzzle in the world. ... In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... Dell Magazines is a company founded by George T. Delacorte Jr. ... A sudoku puzzle Sudoku ), also known as Number Place or Nanpure, is a logic-based placement puzzle. ... Nikoli (ニコリ) is a Japanese publisher that specializes in games and, especially, logic puzzles. ... Kappa Publishing Group, Inc. ... GAMES Magazine (ISSN 0199-9788) is a United States-based magazine devoted to games and puzzles, and is published by GAMES Publications, a division of Kappa Publishing Group. ... The Guardian is a British newspaper owned by the Guardian Media Group. ...

Standard play and terminology

The canonical Cross Sums puzzle is played in a grid of filled and empty cells — "black" and "white", respectively — usually 16×16 in size but can vary widely. Apart from the top row and leftmost column — which are entirely black — the grid, just like a crossword, is divided into "entries" — orthogonal lines of white cells — by the black cells. The black cells themselves — possibly barring those in a cluster — are not entirely solid but rather 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 in the black half-cell immediately above it. These numbers, continuing the borrowed crossword terminology, are commonly called "clues". Canonical is an adjective derived from canon. ...

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 Cross Sums with unique solutions possible, and which means solving a Cross Sums involves investigating combinations more, compared to sudoku in which the focus is on permutations. It has been suggested that Permutations and combinations be merged into this article or section. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

Some publishers prefer to print their Cross Sums grids exactly like crossword grids, with no labelling 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 (at least, not beyond the degree of needing to look outside the grid to solve).

In discussing Cross Sums 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 called as such, despite the "-in-two" and "-in-eight" being equally implied.

Solving techniques

Although brute-force guessing is of course employable, a better weapon is the understanding of the various combinatorial forms that entries can take for various pairings of clues and entry lengths. Those entries with sufficiently large or small clues for their length will have fewer possible combinations to consider, and by comparing them with entries that cross them, the proper permutation — or part of it — can be derived. The simplest example is where a 3-in-two crosses a 4-in-two: the 3-in-two must consist of '1' and '2' in some order; the 4-in-two (since '2' cannot be duplicated) must consist of '1' and '3' in some order. Therefore, their intersection must be '1', the only digit they have in common.

A "box technique" can also be applied on occasion, when the geometry of the unfilled white cells at any given stage of solving lends itself to it: by summing the clues for a series of horizontal entries (subtracting out the values of any digits already added to those entries) and subtracting the clues for a mostly-overlapping series of vertical entries, the difference can reveal the value of a partial entry, often a single cell.

It is common practice to mark potential values for cells in the cell corners until all but one have been proven impossible; for particularly challenging puzzles, sometimes entire ranges of values for cells are noted by solvers in the hope of eventually finding sufficient constraints to those ranges from crossing entries to be able to narrow the ranges to single values.

Some solvers also use graph paper to try various digit combinations before writing them into the puzzle grid. Graph paper is paper that is printed with fine lines making up a grid. ...

Possible sums

Here is a list of some of the clue/length pairings with only one legal combination in a Cross Sums puzzle; note that the order of the digits must still be determined:

3-in-two: 1, 2
4-in-two: 1, 3
16-in-two: 7, 9
17-in-two: 8, 9
6-in-three: 1, 2, 3
7-in-three: 1, 2, 4
23-in-three: 6, 8, 9
24-in-three: 7, 8, 9
10-in-four: 1, 2, 3, 4
11-in-four: 1, 2, 3, 5
29-in-four: 5, 7, 8, 9
30-in-four: 6, 7, 8, 9
15-in-five: 1, 2, 3, 4, 5
16-in-five: 1, 2, 3, 4, 6
34-in-five: 4, 6, 7, 8, 9
35-in-five: 5, 6, 7, 8, 9
21-in-six: 1, 2, 3, 4, 5, 6
22-in-six: 1, 2, 3, 4, 5, 7
38-in-six: 3, 5, 6, 7, 8, 9
39-in-six: 4, 5, 6, 7, 8, 9
28-in-seven: 1, 2, 3, 4, 5, 6, 7
29-in-seven: 1, 2, 3, 4, 5, 6, 8
41-in-seven: 2, 4, 5, 6, 7, 8, 9
42-in-seven: 3, 4, 5, 6, 7, 8, 9
45-in-nine: 1, 2, 3, 4, 5, 6, 7, 8, 9

Any eight- or nine-cell entry has only one combination: nine-cell entries always have all digits from '1' to '9' and therefore are always clued as "45"; eight-cell entries are necessarily clued as 45 minus the value of the missing digit. Kakuro Combinations is a complete table of combinations while Kakuro Helper by Koalog is an interactive assistant displaying the combinations corresponding to a given sum and a given number of cells (These are also candidates for solving the puzzle known as Killer Sudoku.) Killer sudoku (also killer su doku, sumdoku, sum doku or samunamupure) is a puzzle that combines elements of sudoku and kakuro. ...

For practical solving an abbreviated list is useful as follows

Minimum values for each clue length:

• 3 = 1, 2
• 6 = 1, 2, 3
• 10 = 1, 2, 3, 4
• 15 = 1, 2, 3, 4, 5
• 21 = 1, 2, 3, 4, 5, 6
• 28 = 1, 2, 3, 4, 5, 6, 7
• 36 = 1, 2, 3, 4, 5, 6, 7, 8
• 45 = 1, 2, 3, 4, 5, 6, 7, 8, 9

(the triangular numbers) A triangular number is a natural number such that the shape of an equilateral triangle can be formed by that number of points. ...

Maximum values for each clue length:

• 17 = 9, 8
• 24 = 9, 8, 7
• 30 = 9, 8, 7, 6
• 35 = 9, 8, 7, 6, 5
• 39 = 9, 8, 7, 6, 5, 4
• 42 = 9, 8, 7, 6, 5, 4, 3
• 44 = 9, 8, 7, 6, 5, 4, 3, 2
• 45 = 9, 8, 7, 6, 5, 4, 3, 2, 1

This list can be constructed rapidly and provides an effective memory aid to the additional unique combinations. For example the minimum value combination for clue length four is 10 = 1, 2, 3, 4, the next lowest value for this clue length is 11 and the only combination which produces this is 1, 2, 3, 5.

Mathematics of Cross Sums

Cross Sums are NP-complete [1]. In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use...

There are two kinds of mathematical symmetry readily identifiable in Cross Sums. Minimum and maximum constraints are duals, as are missing and required values.

All sum combinations can be represented using a bitmapped representation. This representation is useful for determining missing and required values using bitwise logic operations. In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ...

Variants

A relatively common variant of Cross Sums is its logical successor, Cross Products (or Cross Multiplication), where the clues are the product of the digits in the entries rather than the sum. Another variant is Arrow Numbers, where the combinations for each clue value cannot be repeated within the grid.

The final puzzle of the 2004 United States qualifier for the World Puzzle Championship is titled Cross Number Sums Place: it is a Cross Sums where every row and column of the grid (except the top row and leftmost column as usual) contains exactly nine white cells, none of which — even across multiple entries — are allowed to use the same digit twice, like a Number Place (Sudoku); in addition, small circles are printed on the borders between some white cells; numerically adjacent digits must be placed astride those circles, and may not appear orthogonally adjacent when not astride a circle. The World Puzzle Championship is an annual international puzzle competition run by the World Puzzle Federation. ... A sudoku puzzle Sudoku ), also known as Number Place or Nanpure, is a logic-based placement puzzle. ...

References

• Tutorial at Nikoli (Macromedia Flash required)
• Cross Sums Number Combination Guide: Guide detailing the various number combinations that are possible for a certain number of blanks and a certain number of spaces. Site also has a free number finder for Cross Sums.
• Complexity and Completeness of Finding Another Solution and its Application to Puzzles: Mathematical reference proving NP-completeness.
• How to play Kakuro: Tutorial about how to play Kakuro puzzles and a step by step example how to solve an easy one.
• Will's Kakuro Masterclass: An introduction to Kakuro for beginners
• Kakuro Tutorial — Basic and Kakuro Tutorial — Challenging: Step by step tutorials provided by KakuroConquest.com
• Kakuro Solver tutorial Solve Kakuro Programmatically.
• How to Solve Kakuro Kakuro Tutorial from Indigo Puzzles
• The New Grid on the Block: The Guardian newspaper's introduction to Kakuro

The Guardian is a British newspaper owned by the Guardian Media Group. ...

External links

Wikimedia Commons has media related to: Kakuro

Image File history File links Commons-logo. ... Wikimedia Commons logo by Reid Beels The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ...

Online puzzles

• Kakurolive.com Play Online Kakuro Puzzles for free (also print).
• Kakuro.com Daily puzzles, information and software to download.
• Daily puzzles and Kakuro Flash Game
• Kakuro Flash Game
• Daily puzzles, information, software and Kakuro Flash Game
• Kakuro-World Play unlimited puzzles online or print out, rules, tips, history, challenge
• Kakuro 3 daily puzzles. Play online or print them out. (English)
• KakuroPlay.com Play Kakuro online
• KakuroConquest.com Online Kakuro player
• PuzzlePlayer.com Daily Online Kakuro puzzle
• Indigo Puzzles Online Kakuro player, free daily puzzles and 'logical advice' Assistant

Tutorials and discussions

• Nikoli Kakuro page (also available in Japanese)
• Strategies for Kakuro Puzzles
• Kakuro and sudoku puzzle discussion forum

Software

• A Kakuro Solver
• Open-source Kakuro solver (windows application)
• Top Kakuro For Palm Os
• Mindgames Kakuro Freeware Kakuro puzzle creator and solver with 20,000 puzzles