Secrets of the Kenken masters: Kenken Hints

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 refer to methods that will only work if one is using pencil and paper.

So I’m going to assume a couple things:

  1. You are using pencil and paper.
  2. You have printed the puzzle large enough to write several numbers in each square.
  3. You fully understand the rules of Kenken and do not need a primer on what the symbols in the blank puzzle mean.
  4. You have worked your way through enough 4×4 puzzles to feel ready to do 6×6’s.


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 #3: Create a consistent code that works for you to define the box’s characteristics.  I put the “label” given by the puzzle maker first.  (E.g. 3/, or 7+), followed by the number of boxes in parentheses (E.g. 7+(3) for a box with three squares), followed by whether dupes are possible (in the cases of elbows, tee-shaped boxes and 4-square boxes;  e.g.  7+(3)D or 7+(3)ND or 7+(4)2D.)   Since 2 square boxes are the simplest and are, by definition, in-line, I just put the standard label with no additional notation (e.g. 6+, 3-, 1-, etc.).


Rule #4:  Develop a system to keep track of what combination are possible in each set of boxes.   The on-line systems just allow 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 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.

For example:



another example:



The only boxes I don’t use this approach on are 1+.  These I just lay out the numbers 1 through 6.

Rule #5:  As in the example above, generally try to keep numbers in order from low to high.  (This isn’t always possible, so it’s useful to recognize patterns where the numbers are reversed.)


Rule #6:  Start by filling in the boxes that have the fewest possible alternatives.   (See “Boxes Determined Immediately” below.


Rule #7: Don’t (except for the fun of it) try to keep track of the sequence of the decisions you make.  It’s almost impossible to retrace your steps if you find out late in the game that you’ve made a mistake.  Start by filling in all the squares that have only two possible values.  Maybe even all the squares that have only two possible pairs of values.


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:

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.   For example, take a painfully hard one:  an 3 square L-shaped 14+, or, by my notation  14+(3)D

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 

No other 2 will work, so next comes 3

3, 5, 6   

Again, we’re at the max, so

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.

I don’t write these into the squares unless there are only two or three (or, at most 4) possible combos, or else the diagram gets too cluttered.  It is good to note, however, that certain combos only work in specific squares so an L-shaped 8+ that’s 1,1,6 has to have the  6 in the corner square.



COMBINATIONS DETERMINED IMMEDIATELY, independent of anything else:

2 square boxes:

2x:  1, 2

3x:  1, 3

4x:  1, 4

5x:  1, 5

24x: 4, 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.   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.  Suppose the grid looks like this:

The vertical 3/ is 1 & 3 or 2 & 6, but the top, horizontal box can be 1 or 3 or 2 or 6, so the vertical only eliminates one possible value, not one combination.   


One conundrum resolved:  The beginning puzzler thinks 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.  E.g.:




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.


More to come, but this should get you a long way toward be a solving the most complex Kenken’s out there!

Next:  Finding “impossible combinations of boxes”