## Kakuro

Kakuro is a puzzle given on a grid of red and white cells. The digits 1 to 9 must be filled into all the white cells so that they satisfy the clues given in some of the red cells. The clues specify the sum of the numbers in the row of successive white cells to the right or the column of successive white cells below. No row or column of successive white cells can have a digit repeated. In other words, we can write $4=1+3$ but not $4=2+2$.

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Related topic: Su Doku

# Kakurosities

### Kakuro's younger brothers

Simpler versions of Kakuro can be obtained by tweaking the first rule. Katrio has the rule that you can use only the digits 1 and 2. Then the only number which has a partition is $3 \left(=1+2\right)$. This is the only clue that you can have. Not only is Katrio a bit of a bore, it is not even unique (unless you decide to allow partitions into single numbers: $1=1$ and $2=2$). Kabin passed away early, mourned by the siblings. What is a puzzle to do when it is born with a rule that says use only 1 and that too not more than once?

Kaquadro allows the digits 1, 2 and 3. The allowed clues are 3, 4, 5 and 6. Simple, you say? But did you realize that Kaquadro puzzles can be as large as you wish? Nevertheless you are right. One can prove that Kaquadro is in P and not in NP.

Kapenta, Kahex, Kasept, Kocho and Kanona are part of the mocktail circuit, although Kapenta cannot shake off a feeling that Kakuro is twice as lively. The older siblings are stratospheric. Even Kundici has not been spotted in public in the memory of any of the youngsters.

### Kakuro's cousins

The cousins of Kakuro are obtained by tweaking the second rule. Twokuro demands that each number be allowed exactly twice in each partition. The clues would then have to be even numbers. My friend Simplicio believes that Twokuro gives itself airs: every Twokuro can be exactly transcribed into a Kakuro. I notice that Sagredo is sitting off in one corner with his laptop and not rising to the bait. Perhaps he thinks Simplicio is wrong again?

### Monotonic Kakuro

Adding extra rules simplifies Kakuro. A third rule could be added: that the digits which solve a clue must be written in increasing order along the row or column. With this rule 3=1+2 is allowed, but not 3=2+1. Call this Ascending Kakuro. One could also have Descending Kakuro, in which the solution must have digits written in decreasing order. Ascending and descending Kakuro are two members of the family of Monotonic Kakuro. A moment's thought will assure you that these are simpler than Kakuro. But how simple? Can you look at a puzzle and decide whether it belongs to Monotonic Kakuro rather than Kakuro?

© Sourendu Gupta. Mail me if you want to reproduce any part of this page. My mailing address is a simple (satisfactory) puzzle for you to solve. Created on 30 November, 2006. Last modified on 4 December, 2006.