*This is part 5 of a "Better Know the USPC" preview series. The United States Puzzle Championship is scheduled for August 27th, 1PM EDT.*

**Type:**Kakuro (aka Cross Sums) Variants

**USPC History:**A mostly standard kakuro/cross sums has appeared four times: 1999 - with three missing clues, 2000, 2001 - with 15's for all down clues, and 2005. The first two are from Puzzelsport, the third from Sidney Kravitz, and the last from Michael Rios.

But the target of this entry is an even more frequent brand of kakuro, the kakuro variation, a high pointer that is close to if not the very last puzzle on the test. If you're solving one of these, it's certainly from Michael Rios who has provided all five of them, including 2002 - Hot Cross Sums (each clue is off by 1), 2004 - Cross Number Sum Place (combination of cross sums, number place, and "consecutive" clues), 2006 - Plus or Minus Kakuro (some cells contain negative values), 2007 - ORu Kakuro (each entry has two possible clues but only one is correct), and 2008 - KakurOh (some cells are larger than 1x1 squares and belong in multiple rows/columns). While we haven't had a kakuro variation in two years, I expect to see one again soon. And if not a kakuro variation exactly, certainly another puzzle where a combination of arithmetic and new thinking are needed together.

**Notation tips:**Not as much to share here, but my main notation style is to mark sure pairs on the edges between cells (like 79 in a two-cell 16 clue) as well as the occasionally either/or placement of a given number (ie one of these cells must be a 5) when it similarly lies in adjacent cells. When a single cell has only two options, that gets two light numbers in the center of the cell and is not confused for the other kind of notation. These notes are rather similar to my sudoku notes, only I make many many more notes when solving a sudoku. Here is a solved grid from the 2004 variation that is pretty representative of how much writing is in one of my grids since I never erase my notes.

**Strategies:**I suppose this section needs two separate parts, on kakuro strategies and on kakuro variation-specific strategies. We'll start with the "regular" style before getting more twisted.

Learning to recognize extremes/outliers in clues is almost always a good approach in any kind of logic puzzle, and this is particularly useful in kakuro where noticing the importance of certain values and lengths is the most significant part of solving these puzzles fast. Indeed, the most common "cheat sheet" people would request in a kakuro is exactly a table of the extreme sum values. Learn this simple rule: the

*two smallest*and

*two largest*possible sums for any length will always have a unique digit breakdown. In two cells, 3 and 4 are the smallest sums and must be {1,2} and {1,3} in some order respectively. Equivalently, 16 and 17 with {7,9} and {8,9} are the unique pairs on the high end. In three cells, the magic values are 6 and 7 on the low end, and 23 and 24 on the high end. This continues. In four cells: 10, 11, 29, 30. In five cells: 15, 16, 34, 35. In six cells: 21, 22, 38, 39. In seven cells: 28, 29, 41, 42. When you get to eight cells, all the sums have a unique missing digit so the information you really want to store in your head is "what is not here" which is the x that satisfies 45-x = clue. Often the missing digit is a key digit in an intersecting clue and could force just one entry, so don't specifically ignore eight-cell clues because they seem long. The only useless clues are the 45's which just indicate that all digits from 1 to 9 appear in that entry and nothing can be eliminated until the crossing clues add context.

But while the unique extremes are key, and should become "automatic" when seen in a puzzle, you'll want to get your head around some of the next larger/smaller values which only have two choices. With n cells in the clue, exactly n-2 of these will be fixed for these two value sets. An 8 in three cells must be 1,3,4 or 1,2,5 so a 1 is always present. A 12 in four cells must be 1,2,4,5 or 1,2,3,6 so a 1 and a 2 are always present. And so on. A 23 in six cells contains 1+2+3+4 and either 5+8 or 6+7. One picture people can have in their heads to grasp this principle is to think of sliders sitting over values on a number line, maybe like an abacus. If all the sliders are on the left, you get the smallest possible value. At this stage, only one slider can move (the largest number) to go to a value one higher. For example, OOOOXXXXX = 1234XXXXX can become OOOXOXXXX = 123X5XXXX. But 1,2,3 cannot move as they are blocked to their immediate right. Having gotten to 11, now either of two sliders can move to get to 12. This is the 3 or the 5 (to give 12X45XXXX or 123XX6XXX). The clue value isn't large enough to allow the sliders trapped on the far left, the 1 or the 2, to move yet.

In addition to the fixed digits in these sets, the other kind of thinking to keep in mind is the largest/smallest number that can go in that clue. A 12 in four cells can never contain a 7, 8, or 9. Moreover, if it did contain a 6, then no other cell can be larger than a 3. So when I look at a 12 clue in four cells, I'm looking at the crossing clues specifically for those spots that can take a 1 or 2, since those must go somewhere, as well as for spots that really want a large digit, since it may be those can only take a 5 or 6 and that will constrain all the rest. And while min/max thinking (looking at small clues and big clues that intersect) will often be how you get started, sometimes just knowing the forced number sets is enough. I've seen way too many 4 in two/7 in three and 16 in two/23 in three crossings and placing the forced digits should be automatic.

Realize that as you write numbers down the remaining clues may now become another min/max set. A 9 sitting in a 20 in five cell clue has left behind an 11 in four which you should know is {1,2,3,5}. And occasionally you'll get an intersection with a unique set that normally isn't. A 6 entered into an 18 in five cells leaves a 12 in four which isn't usually unique except that the 6 eliminates all but the {1,2,4,5} option. Every digit placed should get you thinking about the leftovers in the crossing clue as these will give the next sure placement the majority of the time.

While the above will get you through 95+% of the kakuro out there, the next most useful things to keep in mind involve sudoku-like eliminations. You'll sometimes find pairs or even X-wings in kakuro puzzles. And this is why I recommend a notation where I mark any number that must go in one of two adjacent cells (and certainly any sure pairs). In an across entry that intersects two marked vertical 98 pairs, for example, an intersection with a 24 in three cells will only have one option left in that cell, a 7, as the 8 and 9 are accounted for elsewhere even if not written down for sure yet. It's harder to keep in mind all three vertical clues that cross in that row but having some notes down will get your eyes focused in the right spots. There are many valuable extensions of this thinking involving uniqueness where you need to avoid deadly patterns. If a vertical 17 in two cells shares two rows with a vertical 24 in three cells, I can tell you that a 7 cannot go in the outlying cell in that 24 or you'd have two solutions. I don't enjoy writing the consequence of this thinking down, but I certainly will write it down to solve just a bit faster.

In a subset of kakuro, almost exclusively computer generated ones, you may also find a region of the grid mostly isolated from the rest of the grid with only one entry connecting it to the rest. In these cases, in that isolated region you can add up all the vertical clues, add up all the horizontal clues, and from the difference in these values figure out the sum of a single cell (sometimes more) in the connecting clue. I don't like doing this much math, but on a site like croco-puzzle I will certainly put in a valid guess for an entire isolated area (even duplicating numbers) to get the value of the singleton cell. I'll then erase the corner if wrong and work back, now from the connecting entry, to figure out what the unique fill is. I'm sure there could be a counter-example of a Nikoli puzzle that needs this kind of logic, but I'd estimate <1-2% of their puzzles have isolated subsections where this kind of thinking, even mid-solution, is very valuable.

So that's a lot to digest, and that's just how to solve a regular kakuro. So I'll be more pithy with kakuro variants, particularly as I cannot predict what particular variant will appear in any given year. To prepare for these variants, you want to engage in a game of leveling with the puzzle designer. Something like "I know you know I know what a 10 in four cells does, but if X, then ...". You should be able to use the guidelines above to predict what "obvious" work-ins will look like in a new variation. For example, in 2006 there was a Plus or Minus Kakuro where a clue shaded in gray needed at least one value of opposite sign. This change negated the value of "minimal" clues, but did not negate the "maximal" clues. What is the largest possible clue (by absolute value) for a five-cell entry? Well, it's 30 (the largest four-cell entry) minus 1 (the smallest one-cell entry) = 29. If you had internalized information like that before the test, you'd immediately see the work-in for that puzzle in the upper right as well as another in the lower right. Always recalibrate your expectations to match the gimmick. If the gimmick is simply to allow 0 in addition to 1 to 9, then recalibrate all the small clues you keep in your head; if the gimmick is to have each entry one off the correct clue, then look for 18's and 17's in two cells instead of 16's and 17's when you are trying to spot unique outliers, and expect a 16 clue to actually be a 15 as the author is probably trying to be "sneaky".

I have previously constructed some "Nonconsecutive Kakuro" where consecutive digits cannot touch. In that variation, you should recognize that an entry like 6 in three cells is impossible (the 2 would be touching a 1 or a 3 wherever it is placed) and an entry like a 7 in three cells must be 142 or 241 as you have to separate the 1 and 2. Try to figure out what ways a 10 in four cells can be filled in. It's many fewer than the usual 4x3x2x1 options. You need to think in sets of digits, and separate possible consecutive neighbors, to make progress.

**Comments:**I like Nikoli Kakuro books specifically to race against the indicated times. The puzzles don't hide a lot of new techniques, and I've memorized most of the patterns indicated above, so I am pretty efficient at them but not at H. Jo level and unlikely to improve.

But I capital-L Love kakuro variants. If I had just a little more energy and time, I would manage to finish and self-publish a book of my ideas in this area called "Mutant Kakuro" to go along with what I've done to sudoku puzzles. I strongly believe that small changes in the rules to a simple puzzle type can lead to fun emergent properties with new logic to discover and kakuro is as obvious and productive a setting for this experimentation as sudoku is. And Dr. Sudoku already has a Kakuro Lad side-kick ready to extend the story. It's just that kakuro is far less commercial. But I'd list some of my existing mutant kakuro as some of my best puzzle work, and my collaboration with Dan Katz in the 2009 Mystery Hunt is the simplest example to point to to take up several hours of your time with a lot of variety.

So, if you know all that about me, it should come as no surprise that I look forward to new kakuro variations on the USPC every year and while the last few years have been disappointing, as we've gotten KenKen and other math puzzles like those X-Agony and Di-Agony challenges instead, I fully expect a fun original Kakuro variation (with strained/punny title) to appear again soon.

**About this puzzle:**This puzzle idea grew out of the most "ahead of its time" variation in Cross Number Sum Place, one year before the sudoku boom but a really great combination of those two types. Removing the sudoku rules but accentuating the consecutive (and implied nonconsecutive) constraints, I figured I'd make a themed USPC Consecutive Kakuro this week and managed a pretty reasonable end puzzle for a USPC, probably a 30-pointer. Enjoy!

**Rules:**

Enter a single digit from 1 to 9 into each empty square so that the sum of the digits in each across and down answer equals the value given to the left or above, respectively. No digit is repeated within a single answer.

Additionally, a gray bar is shown between two squares every time they contain consecutive digits and only when they contain consecutive digits. (Consequently, any two squares without a gray bar between them cannot contain consecutive digits.)

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