This article is written for those who want to solve the puzzles themselves and figure out their own mechanisms, etc., but would like to get some basic concepts clear that will lead to a smoother process of solving the puzzles. It is designed for a person who likes to solve the puzzles on paper, though most of the discussion is appropriate even if one can keep everything in one’s head — something I definitely cannot do. Some of the hints, however, refer to methods that will only work if one is using pencil and paper.
So I’m going to assume a couple things:
- You are using pencil and paper.
- You have printed the puzzle large enough to write several numbers in each square.
- You fully understand the rules of Kenken and do not need a primer on what the symbols in the blank puzzle mean.
- You have worked your way through enough 4×4 puzzles to feel ready to do 6×6’s or larger puzzles.
Basic vocabulary: A square is the smallest unit of a grid. So there are 36 squares in a 6×6 grid. A box is a set of 2 or more squares identified by an arithmetical operation.
Rule #1: Have a good eraser. Mistakes will happen. If you don’t discover them ’til close to the end of the puzzle, just reprint the grid and start over.
Rule #2: Double check the arithmetical operation and your options. Incorrectly seeing a division sign as a + or a – sign as a plus is very easy to do. For the same reason, double check your entries to make sure you haven’t just simply put the wrong numbers in. I can’t tell you how many times I’ve found I’ve had 2 + 6 = 9 or some such stupidity.
Rule #2A: If you don’t immediately know what combinations are possible for a given box, take the time to figure out the options. (If more than 3 or 4 options are possible, make a note and come back to the box at a later point. If you’ve gotten this far, you probably know some common box possibilities and have noticed some special properties. For example, a 2 box multiplication box equalling 48 can only be the 6 & 8 pairing. My first time through, I’ll sometimes put a dot in large addition squares just to remind myself that each square could be almost anything. These are usually the last to fill. Trying to solve them too early often leads to more errors than solutions.
Rule #3: Work out a consistent code that works for you to made accurate notes outside the grid as you work out details. I particularly make a note on a complex box if it is in a line or has an elbow. I use ND to remind myself that no duplicates are possible and D to remind myself this box can have dupes. I see commas to separate values that are definitely necessary in a given box and the | sign to remind myself that it may a given pair or more of numbers OR a different set.
Rule #4: Develop a system to keep track of what combination are possible in each set of boxes. Most on-line solving just allows you to keep track of what you’ve eliminated. I find this inadequate since you really want to keep track of possible combos, particularly possible pairs, not just which of the numbers from 1 to 6 or more have been eliminated. (Some people can keep track of this stuff in their heads: I find that amazing!) It is very important to line things up so that they help you visually. For consistency, I keep the alternative possible combinations stacked vertically.
Rule #5: As in the instance above, generally try to keep numbers in order from low to high. So I’l for a 12x, I’d put 2 & 6 on the left hand side of the boxes and 3 & 4 on the right. (This isn’t always possible, but it does help to see what’s going on when you can.)
Rule #6: Start by filling in the boxes that have the fewest possible alternatives. (See “Boxes Determined Immediately” below.) I usually begin by ONLY entering the boxes that have only ONE possible pairing (or set of three in a few instances). Then enter (I usually do so in a lighter pencil stroke) the squares that have only TWO possible solutions. A common one is 24x in a 8×8 or 9×9 puzzle. 3 & 8 and 4 & 6 are the only possible combinations that can work in a 2 square box. Gradually, more and more of these “automatic” combos will get stuck in your head.
Rule #7: See if you can work out a system to track the decisions you make. Generally speaking, using lines and arrows does NOT work, because the diagram gets hopelessly complex. I tried numbering my choices, but then there are too many numbers all over the place = more errors. I’ve been working on a system using letters in alphabetic order to show me what the precedents for a given choice were. I put the letter in a corner to once I know which pair (or set or 3 or more) is the only possible solution at a given point in solving. I put a circle around a number when it’s the ONLY possible number in a given box. This system, with a few tweaks for more complex situation, allows me to go back to the beginning and catch where I made a mistake. This can be a fun challenge all it’s own. Or, because, after all, it’s just a puzzle and there are millions more out there, you can always just throw it in the trash and start a new one!
Rule #8: Figure out, without writing them in the squares, all the possible values of arithmetical operations you haven’t seen before or aren’t sure of. This is the method I’ve used successfully, though I’m sure there are other surefire approaches:
Method to determine which combinations are possible:
(Remember these are not done in the boxes themselves but in a space reserved for them outside of the grid.
Step 1: Be clear whether one or more duplicates are possible. L-shaped and T-shaped squares can have one possible duplicated value; 4-square shaped and S-shaped boxes can have 2 duplicated values.
Start with the the lowest value that can work. And gradually go up. The second number must always be equal to if dupes are possible, or, higher than the first if the boxes are all in a line.
For example take a 3 square L-shaped 14+ (by my notation 14+(3D) in 6×6 puzzle.
1 will NOT work, but 2 will, so first I put the 2, followed by a comma
2 will work with 6 & 6 (remember, this is L-shaped so one dupe is OK) so 2, 6, 6 is the only solution with a 2.
3 will work: 3, 5, 6
4, 4, 6 (Important note: as you go up, the second number MUST be equal to the first or higher or else you’re repeating something you should have tried with the lower number.
continuing up, 4, 5, 5 also works
5, 5, 5 exceeds 14, so we’re done with possibilities.
COMBINATIONS DETERMINED IMMEDIATELY, independent of anything else (in a 6×6 puzzle):
2 square boxes:
2x: 1, 2
3x: 1, 3
4x: 1, 4
5x: 1, 5
24x: 4, 6
5-: 1,6
11+: 5,6
3 squares, in-line:
6+: 1, 2, 3
36x: 2, 3, 6
There are many more “automatics.” Keep track of them as you discover them.
It’s also good to keep track of boxes that have only 2 possible pairs. For example, in a 6×6 puzzle, 9+ can only be 3 & 6 or 4 & 5. 3/ can only be 1 & 3 or 2 & 6, and so on.
Some common 3 pair combos often appear frequently enough you get used to seeing them. Often, if you can eliminate 1 of the 3 pairs as being impossible, the 2 pairs left will be a big help in solving the puzzle.
Rule #9: Check for rows or columns that have only one possible 5. In larger puzzles, also check for 7’s and 9s. Sometimes you can find these immediately. If a series of 3/ and 2/ or 5+ cross a row or column, then the remaining 1-, for example, must have the 5 in it. Sometimes this isn’t the case initially, but after you’ve eliminated some possibles, you often find there’s only one place the 5 can be. Later in solving the puzzle, I often go through all the rows and columns to see if there’s any number that can only be in one square (or must be in a certain box). This is often critical to solving the puzzle. Only rarely does one find that three squares in different boxes (it’s common in a single box; e.g. 6+(3)ND = 1,2,3) contain only 3 possible numbers as frequently happens in Sudoku, but it’s still worth looking for.
Rule #10: Find the hidden impossibles! This is really the core of the game once you’ve dealt with the preliminaries. The simplest example is when an equation that has only two pairs appears in 2 boxes in line with each other, e.g. 3/ on top of a 3/. Since each box can only be either 1 & 3 or 2 & 6, between the two boxes all 4 numbers must appear so the remaining box in a 6×6 puzzle must be 4 & 5. But beware: This is also the biggest trap and easiest mistake to make.
One conundrum resolved: At first a puzzler may think that you can’t have identical pairs parallel to each other, but then realizes that the numbers could be reversed in each pair of squares. For example, you could have 4+ parallel to another 4+ box and have 1 and 3 in the first pair and then 3 and 1 in the one parallel to it. It turns out you were right the first time. It took me forever to figure it out why, but you never see this kind of parallel reversal occur in a real puzzle. Why? Not because it’d be wrong from a solver’s perspective, but because it’s wrong from a puzzle builder’s perspective. If this did occur, there’d be two alternative valid solutions, a no no in Kenken world, because nothing in either the row or the column could determine which was right. So if you have two parallel pairs that would work if the values were reversed, you can be sure that one of the two pairs must have a different solution, in this case, one must be 2 & 6.
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I am a 78 year old woman who, before age 77, had never before been drawn to any form of puzzles . My puzzle happening occured when, one day, as I was reading the Arts section of the NYTimes, I accidently allowed my eyes to pause for a moment on the puzzle page. For a moment I wondered what others found intriguing and so I made my first attempt at a 4X4 kenken. It was fun and I was rewarded with my success. Little did I realize, at that moment, how this experience would change my daily routine, I was hooked, my morning coffee now is not complete without kenken. It took my a while to feel comfortable with my success or lack there of. Now I am able to solve 6×6’s with reasonable effort, however, your 7X7’s require much more work resulting in frustration where success is possible only with great effort. If I walk away in frustration I will NOT return to that particular 7X7 puzzle , instead I will await my next Sunday Times challenge. Thank you for adding kenken to my fitness routine of daily water aerobics for my heart and daily ken ken for my brain. The aging Mermaid
I do 9×9 no-op expert puzzles all the time. Is there a harder Ken-ken puzzle available?
I find some of the Inky 9×9’s quite challenging (https://krazydad.com/inkies/). I imagine there could be something bigger, but instead I got interested in developing a system to track the decisions I make as I go along. That proved to be an interesting challenge.
One of the truly awesome solving techniques for diabolical puzzles, lies in realizing that any row or column must add up to the same number. Let’s call it the “primary grid sum” For example, a 6×6 row or column must add up to 21 ( This also uses an old math hack to determine the sum of a series, namely an x grid will sum up to (x*(x+1))/2 or (6*7)/3 = 21).
If you know the only possibilities in several squares, you can subtract their sum from the “primary grid sum” to then limit the possibilities in the remaining, unsolved squares by applying hidden impossibity logic.
Great idea.
You can also use a related approach on groups of adjacent lines (a term I use for rows or columns). If you know the parity of a box (know if the sum of all the numbers in a box is odd or even), you can determine the parity of a single unknown box in the group of lines. This can often help reduce choices or absolutely determine the box’s values.
If the sum of the numbers in a row is even (sizes 4, 7 and 8), then for all adjacent combinations of rows and columns, the final parity must be even. If it is odd (sizes, 5, 6 and 9) then an even number of adjacent rows or column will have an even parity. An odd number of adjacent rows or columns will have an odd parity.
Sometimes the lines (rows or columns) don’t line up smoothly, with a square sticking out. If we know that square’s parity, we can adjust accordingly and still use this method.
Dr O,
I love my 9×8 kenkens but I can’t find good explanations of parity. are you able to direct me please.
B
Well, let’s say your 9×8 means 9 horizontal rows and 8 vertical columns. All rows or groups of rows will have “odd” parity, and all groups of columns will be even.
I hope that helps.
I think I have a good corollary to the strategy. It is handy to know if the sum of a row is even or odd. For instance, a 7X7 adds up to 28, an even number. Following that, any number of successive rows (or cols) would also add up to an even number. Any sums that you have in that row are clearly even or odd. The neat trick is you can also tell that for differences, although you don’t know what those two numbers add up to. For instance, the numbers satisfying 4- will always sum to an even number, and the numbers satisfying five minus will always some to an odd number. Even minus sums to even and odd minus sums to odd. This is help me many times illuminate some possible numbers.
If each row sums to an odd number, clearly two rows will sum to even, three to odd, etc.
Looks clever. I’ll have to try it.
Thanks for your hints. Am just beginning my KenKen study after reaching my limit at Sudoku where I had developed my own tool that normally worked after having gone though the various wings and fish techniques. Was a couple years ago and I couldn’t find enough truly nasty puzzles. So when I heard the term “sudoku on steroids” it was time to begin my studies. Thanks for your tips especially confirming there should be but one solution. That underlies so many sudoku techniques too as you certainly know.
Cheers from Princeton, NJ