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Kakuro or Kakkuro or Kakoro (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 many math-and-logic puzzle publications across the world. In 1966,[1] Canadian Jacob E. Funk, an employee of Dell Magazines, came up with the original English name Cross Sums [2] 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.[2] 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 half-cell to its immediate left and each vertical entry has a number in the half-cell immediately above it. These numbers, borrowing crossword terminology, are commonly called "clues". The objective of the puzzle is to insert a digit from 1 to 9 inclusive into each white cell so 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. Like Sudoku, solving a Kakuro puzzle involves investigating combinations and permutations. There is an unwritten rule for making Kakuro puzzles that each clue must have at least two numbers that add up to it, since including only one number is mathematically trivial when solving Kakuro puzzles. At least one publisher[3] 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 either the solution nor the logic required for 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, both "43-in-eight" and "44-in- eight" are still frequently called as such, despite the "-in-eight" suffix being equally implied. Solving techniques
Combinatoric techniques Although brute-force guessing is possible, a more efficient approach is the understanding of the various combinatorial forms that entries can take for various pairings of clues and entry lengths. The solution space can be reduced by resolving allowable intersections of horizontal and vertical sums, or by considering necessary or missing values. 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. When solving longer sums there are additional ways to find clues to locating the correct digits. One such method would be to note where a few squares together share possible values thereby eliminating the possibility that other squares in that sum could have those values. For instance, if two 4-in-two clues cross with a longer sum, then the 1 and 3 in the solution must be in those two squares and those digits cannot be used elsewhere in that sum.[4] When solving sums that have a limited number of solution sets then that can lead to useful clues. For instance, a 30-in-seven sum only has two solution sets: {1,2,3,4,5,6,9} and {1,2,3,4,5,7,8}. If one of the squares in that sum can only take on the values of {8,9} (if the crossing clue is a 17-in-two sum, for example) then that not only becomes an indicator of which solution set fits this sum, it eliminates the possibility of any other digit in the sum being either of those two values, even before determining which of the two values fits in that square. Another useful approach in more complex puzzles is to identify which square a digit goes in by eliminating other locations within the sum. If all of the crossing clues of a sum have many possible values, but it can be determined that there is only one square that could have a particular value which the sum in question must have, then whatever other possible values the crossing sum would allow, that intersection must be the isolated value. For example, a 36-in-eight sum must contain all digits except 9. If only one of the squares could take on the value of 2 then that must be the answer for that square. Box technique 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. This technique works because addition is both associative and commutative.
"Sudoku From Denksport". Keesing Group B.V. Retrieved November 18, 2018. "Kakuro rules". Retrieved November 18, 2018. Berthier, Denis (April 5, 2013). "Pattern-Based Constraint Satisfaction and Logic Puzzles". arXiv:1304.1628 [cs.AI]. Takahiro, Seta (February 5, 2002). "The complexities of puzzles, cross sum and their another solution problems (ASP)" (PDF). Retrieved November 18, 2018. Yato, Takayuki; Seta, Takahiro (2004). "Complexity and Completeness of Finding Another Solution and Its Application to Puzzles" (PDF). Retrieved November 18, 2018. {{cite journal}}: Cite journal requires |journal= (help) "What is Kakuro". Retrieved November 18, 2018. "Kakuro History". Retrieved November 18, 2018. External links Wikimedia Commons has media related to Kakuro. The New Grid on the Block: The Guardian newspaper's introduction to Kakuro IAENG report on Kakuro Solve Kakuro puzzles online Authority control databases: National Edit this at Wikidata IsraelCzech Republic Categories: Logic puzzlesNP-complete problems1966 introductions This page was last edited on 28 October 2023, at 17:07 (UTC). Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. xxxHolic: Kei Article Talk Read Edit View history Tools From Wikipedia, the free encyclopedia
xxxHolic: Kei No. of episodes 13 Release Original network TBS Original release April 3 – June 26, 2008 Season chronology ← Previous Season 1 List of episodes The second season of xxxHolic, xxxHolic: Kei (xxxHOLiC◆継, Horikku: Kei, literally "xxxHolic: Sequel"), is an anime adaptation of a manga series written by Clamp. It was developed by Production I.G with the main staff and cast remaining the same as in the first season.[1][2] The season aired on TBS on April 3, 2008 in Japan and ended on June 26, 2008, with 13 episodes in total.[3][4] The opening theme is "NOBODY KNOWS" by Shikao Suga and the ending theme is "Honey Honey" by Seamo. Episode list No. overall No. in season Title Original air date 25 1 "Spider (Spiderweb)" Transliteration: "Kumo no Su" (Japanese: 蜘蛛 クモノス) April 3, 2008 Watanuki meets Doumeki at the temple where he lives. While helping Doumeki clean the yard, Watanuki damages a large spider web hanging from a tree. Doumeki clears the spider web away. The next day, Doumeki cannot open his right eye, which is covered in a substance similar to a spider web. Watanuki turns to Yūko, who explains that the spider, whose web Doumeki cleared away, has taken its revenge by taking something of similar value, Doumeki's eye. Watanuki feels that the spider's grudge should be directed at him, even when Yuuko points out that grudges aren't reasonable. That night, Doumeki goes outside and is engulfed by a strong wind which removes the web restoring his right eye. The next morning, Watanuki comes to school with a bandage over his right eye. When Doumeki later rips it off, it is shown that Watanuki's right eye has become blank and lost its vision which angers Doumeki. After school, Watanuki stops at the temple to deliver a book for Himawari and sees that Doumeki is carrying a large number of old books about reversing spells and lifting curses. When he tells Yuuko, she says that Doumeki is trying to lift the spider's curse himself.
While on a picnic with Himawari and Doumeki, Watanuki meets Kohane Tsuyuri, the little girl who is a famous medium and has appeared on TV. Her mother warns Watanuki to stay away from her, but the girl senses that they have something in common. Yuuko tells Watanuki that she is genuine and that they have similar abilities. The next day Doumeki tells Watanuki that he saw the girl with the eye he gave to him. Watanuki sees her again at the cherry blossom tree in the temple grounds and they both see the ghost of a woman who haunts the tree. They understand that the ghost is sustaining the old tree and keeping it alive. They decide to help and lead her to Doumeki's temple. They agree to become friends and part. Kohane goes home, but her relationship with her mother is strained as her mother is exploiting Kohane’s abilities for money. 30 6 "Peace (Konohana)" Transliteration: "Konohana" (Japanese: 平和 コノハナ) May 8, 2008 At Doumeki's place, Watanuki plays mahjong with Yūko, Mokona, and Doumeki. Watanuki loses badly because he doesn’t know how to play. Ame-warashi and Zashiki-warashi arrive and try to help Watanuki, but he still loses. As the game comes to an end, the spirit of the woman who haunted the cherry tree smiles and departs. Yūko tells Doumeki that for some people, the sound of clacking mahjong is said to bring good fortune. 31 7 "Water Cat (Water-Drawing)" Transliteration: "Mizukumi" (Japanese: 水猫 ミズクミ) May 15, 2008 The Neko-Musume (cat-girl) asks Yūko to lend her Watanuki for a job: Watanuki is to draw water from a well in exchange for an egg. To access the well, Watanuki is forced to trespass into a private garden of a house and notices a woman sitting motionless at a high window. When he returns for more water with Doumeki, the woman’s position is unchanged. Watanuki establishes that the woman is not a spirit, because Doumeki can see her with his normal eye. Later, Watanuki 'pinky- promises' Himawari that he won't overwork himself. When he next collects water, his little finger hurts and the wrapper of his water container is blown into the window of the woman in the house. Doumeki and Watanuki enter the empty and silent house. Doumeki and Watanuki find the woman, who falls inert to the floor in a gust of wind. Watanuki assumes she's a doll. Yūko appears and asks Doumeki who says that it is a corpse. Watanuki protests that a corpse would have putrefied in the summer heat, but Doumeki points out that, according to his grandfather, corpses of the modern age do not rot because of the preservatives people consume. Meanwhile the Neko-Musume retrieves a dead kitten from the almost-emptied well. As she leaves, she comments on the miasma around Watanuki's little finger. This indicates that something is strange about Himawari. 32 8 "Ring (Whisper)" Transliteration: "Sasayaki" (Japanese: 鈴音 ササヤキ) May 22, 2008 A girl claiming her house is filled with scary and strange noises seeks Yūko's aid to remove her fear of the house. Yūko gives her a bell to ring, but it makes the noises worse. Two bells have a similar effect. Yūko gives her more and more bells, but the girl says they don’t work. Watanuki is concerned but Yūko says the problem will end that night. Later, in a dream, Watanuki is visited by Doumeki’s grandfather, Haruka, and together they go to the girl’s house. There they observe both a young couple and the girl who cannot see each other, but hear the noise of each other’s presence. Watanuki realises that the girl is a ghost. He then hears that she will be exorcised by Kohane Tsuyuri, and be forced to leave the house. This will fulfil her wish as Yūko predicted, but not in the way he originally expected.
33 9 "Rumor" Transliteration: "Fūhyō" (Japanese: 流噂 フウヒョウ) May 29, 2008 After learning that Kohane is being harassed by the public as many believe her powers are a fraud, Watanuki and Doumeki pay a visit to her with a basket of food. The house has been covered with graffiti. Kohane appears, but is injured from a fall at the TV station where she was pushed down stairs. While eating, Kohane's mother arrives home, in an angry mood and throws hot tea at Watanuki, burning him. Kohane's mother forces her to go on another TV show with other spiritual seers. All the other seers see the same vision of the ghost of a soldier in a haunted house, but Kohane see a woman too. The other seers reject her vision, and she is criticised by the public. 34 10 "No Return (Awareness)" Transliteration: "Kizuki" (Japanese: 不戻 キヅキ) June 5, 2008 Tired of seeing Kohane being harassed by the media and mistreated by her own mother, Watanuki takes her to the shop. She spends a day with them but returns home the next day. Kohane releases the female spirit from the haunted house, and human remains are found, confirming her vision. She becomes respected again by the fickle media and public, however her mother announces Kohane's retirement and they leave their house. Some time later Watanuki sees Kohane at the tree where they first met, and she reassures him the she is well, and her mother is no longer cross with her. 35 11 "Secret (Alone)" Transliteration: "Hitori" (Japanese: 秘事 ヒトリ) June 12, 2008 Watanuki's little finger continues to hurt and he becomes more suspicious of Himawari. In the following day, he drops and breaks a cup because of his finger. Yūko says that Himawari may not be his lady luck. That night he meets Doumeki’s grandfather, Haruka in a dream and discusses the situation with him. Haruka says to stay with Doumeki. The next day Himawari taps him on the back and he accidentally falls out of the window at school when a window gives way. While unconscious, he starts to follow his dead parents, but Haruka pulls him back. A messenger from Yūko takes him to the shop where he eventually regains consciousness. Himawari is there and, before he finishes asking her about the accidents, she reveals that she already knows about them. 36 12 "Truth" Transliteration: "Hontō" (Japanese: 真実 ホントウ) June 19, 2008 Himawari reveals to Watanuki that even though she is human, accidents can happen to anyone around her except her parents. She realises that she should never see Watanuki again, but he resists a farewell. They agree to see each other soon. Yūko tells him that the price to remove bad luck from Himawari is too high to pay. She reveals that Doumeki and Himawari paid the price for saving his life and because of their wishes, they have become her clients. Yūko will not tell him the price they paid
- Doumeki gave his blood and Himawari took his scars. Yūko gives Watanuki the egg given to her by Neko-Musume from which hatches a little yellow bird. Yūko says that if he gives it to Himawari, that no harm will befall it and she can finally have a close friend without fear of bad luck affecting them. 37 13 "Repayment" Transliteration: "O-kaeshi" (Japanese: 報恩 オカエシ) June 26, 2008
Example of a Hitori puzzle
For other uses, see Hitori (disambiguation).
Hitori (Japanese: "Alone" or "one person"; ひとりにしてくれ Hitori ni shite kure ; literally
"leave me alone") is a type of logic puzzle published by Nikoli.
Hitori is NP complete.[1]
Rules[edit]
The same Hitori puzzle, now solved
Hitori is played with a grid of squares or cells, with each cell initially containing a number.
The game is played by eliminating squares/numbers and this is done by blacking them out.
The objective is to transform the grid to a state wherein all three following rules are true:
• no row or column can have more than one occurrence of any given number
• black cells cannot be horizontally or vertically adjacent, although they can be diagonal to
one another.
• the remaining numbered cells must be all connected to each other, horizontally or
vertically.
Solving techniques[edit]
• Once it is determined that a cell cannot be black, some players find it useful to circle the
number, as it makes the puzzle easier to read as the solution progresses. Below we
assume that this convention is followed.
• When it is determined that a cell must be black, all orthogonally adjacent cells cannot be
black and so can be circled.
• If a cell has been circled to show that it cannot be black, any cells containing the same
number in that row and column must be black.
• If blacking out a cell would cause a connected non-black area to become separated into
several unconnected components, the cell cannot be black and so can be circled.
• In a sequence of three identical, adjacent numbers, the centre number cannot be black
and the cells on either side must be black. The reason is that if one of the end numbers
remains non-black this would result in either two adjacent black cells or two cells with
the same number in the same row or column, neither of which are allowed. (This is a
special case of the next item.)
• In case of two identical, adjacent numbers, if another cell occurs in the same row or
column containing the same number, the latter cell must be black. Otherwise, if it
remains non-black, this would result in either two cells with the same number in the
same row or column, or two adjacent black cells, neither of which are allowed.
• Any number that has two identical numbers on opposite sides of itself cannot be black,
because one of the two identical numbers must be black, and it cannot be adjacent to
another black cell.
• When two pairs of identical numbers are in a two by two square on the grid, two of
them must be black along a diagonal. There are only two possible combinations, and it is
sometimes possible to decide which is correct by determining if one variation will cut
non-black squares off from the remainder of the grid.
• When two pairs of identical numbers form a square in the corner of a grid, the corner
square and the one diagonally opposite must be black. The alternative would leave the
corner square isolated from the other non-black numbers.
History[edit]
Hitori is an original puzzle of Nikoli; it first appeared in Puzzle Communication Nikoli in issue
#29 (March 1990).
In media[edit]
• Episode 11 of xxxHolic: Kei is titled "Hitori" in reference to this.
See also[edit]
• List of Nikoli puzzle types
• Kakuro
• Sudoku
References[edit]
1. ^ Hearn, Robert A.; Demaine, Erik D. (2009), "Section 9.2: Hitori", Games, Puzzles, and Computation, A K Peters, pp. 112 – 115
• Puzzle Cyclopedia, Nikoli, 2004. ISBN.
External links[edit]
• Sample Hitori puzzles on the Nikoli web site
• Hitori tutorials on the Nikoli website
