Introduction

KenKen is an exercise in logic and arithmetic. No other skills are required.

The correct solution will have no duplicate values in any row or column and use the digits starting at one.

The number of digits used depends on the dimensions of the grid; e.g., a 6x6 grid uses the numbers 1 through 6.

Equations

A grid is partitioned into regions which we will call 'equations'.

The four equation types are:

- Sum having a symbol of +
- Difference having a symbol of -
- Product having a symbol of × and
- Quotient having a symbol of ÷

Sum (+)

Any number of cells can participate in a sum and the cells can form any shape. Any of the addend values may be assigned to any of the sum's cell.

Difference (-)

Only two adjacent cells can participate in a difference; either cell can solve to the larger value.

Product (×)

Any number of cells can participate in a product; any of the factors can be assigned to any of the product's cells.

Quotient (÷)

Only two adjacent cells can participate in a quotient; either cell can be assigned the numerator value.

There are a number of simple logic ideas that allow you to proceed to a solution.

Candidates

The solution process involves assigning candidates for each cell and diminishing the candidate list as the solution evolves
through the application of arithmetic and logic. Note that the candidates for a particular cell are always written lowest-to-highest:
i.e., in a pair of 2÷ cells of a 6x6 puzzle, the candidates are written as 1 2 3 4 6 even though they represent 3 distinct solutions, namely 1 2, 2 4 & 3 6.

Naked Pairs

In an equation having two cells, a naked pair occurs if the two cells can only be assigned a pair of values.

As an example, if two cells require a product of 15, then one of the cells will solve to 3 an the other will solve to 5.
The value of discovering a naked pair is that the other cells in the row or column in which the naked pair is found cannot be assigned the naked pair values.

Naked Trios

In an equation having three cells, a naked trio occurs if the three cells can only be assigned a trio of values.

As an example, if three cells require a product of 15, the one of the cells will solve to 1, another will solve to 3 and another will solve to 5.
Assuming that the cells in the trio are in the same row or column, the naked trio's values cannot be assigned to the other cells of the row or column.

Orphans

When examining the candidates of the cells in a row or column, an orphan occurs when only one candidate list contains a certain value.

As an example, if the candidate list for a row of a 6x6 puzzle is: a. 1 2 4 5; b. 1 2 3 4; c. 2 3 4; d. 1 3 5; e. 1 4 6; f. 2 3 5; then the fifth cell (e)
can be assigned the value 6 because it is not a candidate elsewhere in this row or column.

Sum-Of-The-Digits Rule

Occasionally, progress is made by applying this rule to a row or column. For instance, in a 6x6 puzzle the sum of the digits in a row or column must equal 21
(for 7x7: 28; for 8x8: 36; and for 9x9: 45). Here is a fairly elaborate example that shows the sum-of-the-digits rule:

The Sum-Of-The-Digits rule will allow you to determine the correct digit-pair for cells E and F. Here is the logic:

- the sum of the eight digits is 36;
- the sum of the digit in A and B is even;
- the sum of the digit in C and D is even;
- the sum of the digit in E and F is odd (11);
- therefore the sum of the digit in G and H must be odd;
- so, only 1 2 and 3 6 are potential solutions for G and H;
- but if the 3/6 pair is chosen, the remaining digits cannot satisfy the three equations of AB, CD and EF;
- Therefore GH solve to 1 2.

| A | B | C | D | E | F |

| 2÷ | 4- | 2÷ | | 15× | 2- |

1 | | | | | | |

| | | 10+ | 9+ | | |

2 | | | | | | |

| 120× | | | | | 6× |

3 | | | | | | |

| | | 2- | | | |

4 | | | | | | |

| 6× | | 1- | 2÷ | 4- | |

5 | | | | | | |

| 3- | | | | 3÷ | |

6 | | | | | | |

Disclaimer

Although each KenKen puzzle has only one solution, there are many solving sequences that will get you there.
The list below is merely one of them.

What to do first

In order for you to learn effectively, do these steps:

- Print the blank puzzle
- get a pencil with eraser and enter each of the solve steps listed below but be sure to understand why before proceeding to the next step:
- [15×] E1 and E2 enter candidates 3 5
- [3÷] E6 enter candidates 1 2 6; F6 enter candidates 2 3 6
- [3-] A6 and B6 enter candidates 1 2 4 5

(cannot be 3 6 because E6/F6 would have no answer)
- [120×] A3, A4 and B4 enter candidates 4 5 6
- [6×] A5 and B5 enter candidates 1 2 3 6
- [4-] B1 and B2 enter candidates 1 2 5 6
- [4-] E5 enter candidates 1 2 6; F5 enter candidates 2 5 6
- [6×] F3 enter candidates 1 2 3;

F4 enter candidates 1 2 3 6;

E4 enter candidates 1 2
- [9+] E3 enter 4 (orphan);

D2 enter 4;

D3 enter 1

(if D2/D3 were 2 3 instead, D5 and D6 would have no answer.
- [120×] A3 enter 5 6
- [6×] F3 enter 2 3; F4 enter 1 2 3
- [2÷] D5 and D6 enter 3 6
- [2÷] D1 enter 2
- [2-] D4 enter 5
- [120×] A4 and B4 enter 4 6
- [120×] A3 enter 5
- [3-] A6 enter 1 2 4
- [3-] B6 enter 1 4 5
- [4-] B1 enter 1 5 6
- [4-] B2 enter 1 2 5

Here are the last steps to the puzzle's solution:

- using your pencil with eraser, continue entering each of the solve steps listed below but be sure to understand why before proceeding to the next step:
- [2divide;] C1 enter 1 4
- [2-] C4 enter 3
- [6×] F4 enter 1 2
- [6×] F3 enter 3
- [10+] B3 enter 2
- [10+] C3 enter 6
- [10+] C2 enter 2
- [1-] C5 and C6 enter 4 5
- [2divide;] C1 enter 1
- [4-] B1 enter 5; B2 enter 1
- [3-] B6 enter 4; A6 enter 1
- [120×] B4 enter 6; A4 enter 4
- [6×] B5 enter 3; A5 enter 2
- [2divide;] D5 enter 6; D6 enter 3
- [1-] C5 enter 4; C6 enter 5
- [4-] E5 enter 1; F5 enter 5
- [6×] E4 enter 2; F4 enter 1
- [3divide;] E6 enter 6; F6 enter 2
- [15×] E1 enter 3; E2 enter 5
- [2divide;] A1 enter 6; A2 enter 3
- [2-] F1 enter 4; F2 enter 6

| A | B | C | D | E | F |

| 2÷ | 4- | 2÷ | | 15× | 2- |

1 | | 1 2 5 6 | | | 3 5 | |

| | | 10+ | 9+ | | |

2 | | 1 2 5 6 | | 4 | 3 5 | |

| 120× | | | | | 6× |

3 | 5 6 | | | 1 | 4 | 1 2 3 |

| | | 2- | | | |

4 | 4 5 6 | 4 5 6 | | | 1 2 | 1 2 3 6 |

| 6× | | 1- | 2÷ | 4- | |

5 | 1 2 3 6 | 1 2 3 6 | | | 1 2 6 | 2 5 6 |

| 3- | | | | 3÷ | |

6 | 1 2 4 5 | 1 2 4 5 | | | 1 2 6 | 2 3 6 |

| A | B | C | D | E | F |

| 2÷ | 4- | 2÷ | | 15× | 2- |

1 | | 1 5 6 | | 2 | 3 5 | |

| | | 10+ | 9+ | | |

2 | | 1 2 5 | | 4 | 3 5 | |

| 120× | | | | | 6× |

3 | 5 | | | 1 | 4 | 2 3 |

| | | 2- | | | |

4 | 4 6 | 4 6 | | 5 | 1 2 | 1 2 3 |

| 6× | | 1- | 2÷ | 4- | |

5 | 1 2 3 6 | 1 2 3 6 | | 3 6 | 1 2 6 | 2 5 6 |

| 3- | | | | 3÷ | |

6 | 1 2 4 | 1 4 5 | | 3 6 | 1 2 6 | 2 3 6 |

| A | B | C | D | E | F |

| 2÷ | 4- | 2÷ | | 15× | 2- |

1 | | 5 | 1 | 2 | 3 5 | |

| | | 10+ | 9+ | | |

2 | | 1 | 2 | 4 | 3 5 | |

| 120× | | | | | 6× |

3 | 5 | 2 | 6 | 1 | 4 | 3 |

| | | 2- | | | |

4 | 4 6 | 4 6 | 3 | 5 | 1 2 | 1 2 |

| 6× | | 1- | 2÷ | 4- | |

5 | 1 2 3 6 | 1 2 3 6 | 45 | 3 6 | 1 2 6 | 2 5 6 |

| 3- | | | | 3÷ | |

6 | 1 2 4 | 1 4 5 | 4 5 | 3 6 | 1 2 6 | 2 3 6 |

| A | B | C | D | E | F |

| 2÷ | 4- | 2÷ | | 15× | 2- |

1 | 6 | 5 | 1 | 2 | 3 | 4 |

| | | 10+ | 9+ | | |

2 | 3 | 1 | 2 | 4 | 5 | 6 |

| 120× | | | | | 6× |

3 | 5 | 2 | 6 | 1 | 4 | 3 |

| | | 2- | | | |

4 | 4 | 6 | 3 | 5 | 2 | 1 |

| 6× | | 1- | 2÷ | 4- | |

5 | 2 | 3 | 4 | 6 | 1 | 5 |

| 3- | | | | 3÷ | |

6 | 1 | 4 | 5 | 3 | 6 | 2 |