Strategies for Deduction

Several different strategies are important for solving all KenKen puzzles using your powers of deduction. We will look at these strategies using examples from the puzzles solved in this blog.

Factoring 

We must be able to break or factor larger numbers down, as shown above, until you get to numbers you can use in KenKen puzzles.  This is useful when the cage you are looking at is one that involves multiplying.

When breaking a number down into its factors it is useful to include 1 and put the factors in order from smallest to largest.  This will help  you to determine in a systematic way what factors could combine together to go in the given cage.



Here is a 10x cage from a 5x5 KenKen Puzzle (Puzzle 6).    We must factor the number 10 down until we get some numbers we can use.    10 = 2x5 = 1x2x5   Note:  it is often helpful to remember that 1 is a factor of every number we will be using, and will sometimes be involved in the solution.   We can see that  1,2,5 must be the numbers that will be found in this cage.  No other factors are available.

Here is a 72x cage from a 6x6 puzzle 
(Puzzle 8).

We factor 72 first by splitting it in two and then factoring those numbers in turn until we reach numbers we can use, which for a 6x6 puzzle would be 1,2,3,4,5,6.

72 = 1x2x36 = 1x2x(2x18) = 1x2x2x2x9 = 1x2x2x2x(3x3). 

Here we have five factors but only three squares in the cage, so we will have to multiply some of the factors together again to get other possibilities for example as follows: 3x(2x2)x(2x3) giving 3, 2x2=4 and 2x3=6 or 3,4,6.  One can quickly see that no other way of combining these factors into three numbers will give us anything that works in a 6x6 puzzle.

Often you will find that there is in fact only one set of numbers that works, so this strategy can be extremely useful.

 Here is a 25x cage from a 6x6 puzzle (Puzzle 8)





We know that 25=1x5x5. If the cage was in a row or column, as with the above two, then we know we would not be able to solve it, since that would involve putting two 5’s in the same row or column. But in this case we are able to use the two 5’s by placing them on a diagonal to one another, and the 1 in the other square. This is clearly the only way to solve this cage. 

Doubles, Triples and Quads

These are techniques that should be familiar to avid Sudoku and Kakuro solvers.

 Here is an example of a column double in a 5x5 puzzle. There are no other number possibilities in a 5x5 problem. For a 6x6 you would also have to look at 2x6. A column double means that you know what are in those two squares and that all the other squares in the column cannot contain either of these numbers. Same works for doubles in a row as opposed to a column.

Here is an example of a column triple from a 6x6 puzzle. We would notice that 15 only has three factors: 1x3x5. Even though we might not know which squares the numbers go in, a triple further narrows down the possibilities for the other squares in the row or column.



Quads are rarer than doubles and triples, but the same principles apply, only for four squares in a row or column.


 Adding by column or row

 Like a Suduko puzzle, all the rows and columns must contain all of the same set of numbers without any duplicates.

This means that if you sum up the squares in any row and column you will get the same total. For a 6x6 puzzle this total would be 1+2+3+4+5+6 = 21.

This information can be used in a variety of powerful ways to sum part of a row or column to deduce numbers from another part:

If we look at this column (from Puzzle 7) we see two 5+ cages which will total 10. The first column number in the 30x cage is 5 or 6. If we add these together we get two possible totals: 15 or 16. Therefore the bottom number in the first column must be a 21-15=6 or 21-16=5, but since 5 is not a possibility there, then the number must be 6









Multiplying by column or row

 



In this example from a 5x5 puzzle (from Puzzle 5) we can see that the numbers in the right hand column must be 1,2,3,4,5. We can therefore figure out the number at the bottom of the left hand column by multiplying together the numbers in the right hand column and dividing this product into 600:

600 / (1x2x3x4x5) = 600/120 = 5.





“Too Big for Your Britches” Strategy

In this 6+ cage (from Puzzle 11) we can see that the bottom number cannot be 5 because 5 is TOO BIG to leave any numbers for the other two squares!







“Little Britches” Strategy

 In the row above (from Puzzle 19, 9x9) we see a variation of the "Too Big For Your Britches" scenario which we will call "Little Britches".  A scan of the row shows that the number 1 is not shown anywhere as a possibility.  We look at the 11+ cage and see that 1 is TOO SMALL to use in that cage, and therefore must go in the empty square in the middle of the row.

 
Scanning Rows and Columns

Sometimes you cannot see the forest for the trees.  When you seem to be at a dead end, it sometimes helps to step back and look at the larger picture:

In scanning the row above from a 5x5 puzzle (Puzzle 5) you suddenly notice that the number 4 does not appear, and therefore MUST be the number in the last, empty square!

Let us now look at the following row from a 7x7 puzzle 
(Puzzle 14).  

We can see that there is no 7 among the possibilities shown.  Therefore we must conclude that 7 is in one of the two empty squares on the left.  If a 7 is in one of these squares then we may also conclude that a 6 must be in the other to solve the 1- cage.

The column here to the left (from 7x7 Puzzle 14) has no 6 showing and so we may conclude that 6 is in one of the two empty bottom squares.

















In Puzzle Number 25 of this blog, we seem to have little to work with at first:

 
But look at the straight line 5 + cage in the second
last row. We know that the 5 cannot be in the 5+ cage and so
must be in one of the other three squares of that row. We know
that all the numbers in the 24x cage must multiply together to get
24, but 5 is not a factor of 24. 5 is a factor of 10,15,20,25,30, etc.
and the pattern is clear and does not include numbers like 24.
There is now only one square left in that row for the 5 and so we have found our first number for this puzzle!

Accentuate the Negative

 Sometimes when possibilities are numerous, it is better to show what is IMPOSSIBLE rather than the possibilities:

For the above row from a 6x6 puzzle (Puzzle 12) we can plainly see that the number 4 is IMPOSSIBLE for the 4- cage.  With this information added, we can scan the row and see that there is indeed only one square in the row that could be a 4.

X-Wing 

This is a strategy that is also used in Sudoku puzzles.  It usually involves having the same number in doubles on two columns or two rows.   Here is an example from  Puzzle 8:

 We notice that in the top two rows there are doubles which both contain the number 4.  This is an X-Wing configuration and means that the number 4 must be in one of these squares in both the fourth and fifth columns.  Therefore we may conclude that no other square in these two columns can contain a 4.  For example we have 2,4 as initial possibilities for the 6x cage in the fifth column, but from the X-Wing we know that the 4 must be in one of the two squares at the top of the column and therefore can be rejected as a possibility in the 6x cage.  Of course we also know that 4 is not a factor of 6, and so we have two separate ways to reject the 4 in the 6x cage.

The above X-Wing example has adjacent squares, but really any doubles on any two columns and rows can be X-Wings.  In the example below (from Puzzle 10) we see a 1,2 double in the top row and a 2,3 double on the bottom row, and together these form an X-Wing that shows 2 can only be in the top or bottom squares of the third and fourth columns.