KenKen Lesson 3: Listing KenKen Combinations
KENKEN RULES
1. Every square in the grid will contain one number.
" In a 5 x 5 puzzle, use the numbers 1 5.
" In a 6 x 6 puzzle, use the numbers 1 6.
2. Do not repeat numbers in any row or column.
3. A cage is a heavily outlined set of squares. The numbers in the squares within a
cage must combine (in any order) to produce the target number in the top
corner. You must use the mathematical operation next to the target number.
4. Cages with just one square should be filled in with the target number in the
top corner. (Hint: This is a good place to start solving the puzzle.)
Puzzle 3A Puzzle 3B
Puzzle 3C Puzzle 3D
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KenKen Lesson 3: Listing KenKen Combinations
OVERVIEW
Students list possible combinations to complete KenKen cages. They also consider the relationship
between combinations and arrangements (permutations).
NCTM STANDARDS
Problem Solving
" Build new mathematical knowledge through problem solving
" Solve problems that arise in mathematics and in other contexts
PROCEDURE
1. Remind students that a KenKen cage is a heavily outlined set of squares with a target number in
the corner. A cage with only one square has only one possible answer: the target number. A cage
with two or more squares must be solved. One part of the KenKen solution process is finding all of
the possible number combinations that match a given target number. In this lesson, students focus on
listing combinations.
2. Point out that some cages have only one possible combination. Show Puzzle 3A to students and
draw attention to the 9+ cage. Ask:
" What is the only possible combination of numbers that can fill this cage? (4,5)
" What equation can you write to show the relationship between the target number and the
solution? (4 + 5 = 9)
" Why is the combination (3,6) not possible? (Only the numbers 1 to 5 are used in a 5 x 5
KenKen puzzle.)
" Can you write the solution in this cage now? (No; you do not know which number, 4 or 5,
goes in each square.)
3. Explain that many cages have more than one possible combination. Draw students attention to the
2÷ cage. Ask: What are the possible combinations of two numbers that can fill this cage? (1,2), (2,4)
Encourage students to write the equations that correspond to each combination (2 ÷ 1 = 2; 4 ÷ 2 = 2)
4. Have students think about how the cage size affects the set of possible combinations by inviting
them to consider the two 8+ cages. Ask:
" What is the only possible combination of numbers that can fill the 8+ cage with two squares?
(3,5)
" What are the four possible combinations that can fill the 8+ cage with three squares? (1,2,5),
(1,3,4), (2,2,4), (2,3,3) Point out that this 3-square cage can repeat numbers because numbers can
repeat on a diagonal, but not in rows or columns.
5. Explain that a combination lists the numbers that might fit in a cage. An arrangement shows the
ways in which those numbers are positioned or placed within the cage. Ask: What is the only
possible combination that can fill the 4+ cage? (1,3) What are the possible arrangements? (1,3) or
(3,1) Point out that mathematicians sometimes use the word permutations to refer to combinations
when the order of numbers is significant.
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6. Tell students that some cages with three squares have only one possible arrangement. If you know
the arrangement, you have solved the cage. Have students look at the 9x cage. Ask: What is the only
possible combination that can fill this cage? (1,3,3) What is the only possible arrangement of these
three numbers? (The 1 goes in the corner.) Why? (The 3 s cannot be placed next to one another
because they would be in the same row or column.)
7. Direct students to list the possible combinations for the remaining cages in Puzzle 3A. Then have
students use this information to help them solve the puzzle. Suggest that students consider the cages
in this order: 4 , 16x, 9+, 3+, 8+ in fifth row, 4+, 8+ in fifth column, 11+, 2÷.
8. Ask students to solve Puzzle 3B., Have them look for a cage that has only one possible combination
and arrangement after they fill in the single square cage. (5x is filled with (1,1,5); the 5 goes in the
corner.)
9. Point out that finding possible combinations helps eliminate possibilities and solve individual
squares. Have students find the combinations for the 6x cage (2,3) and the 15x cage (1,3,5). Then
ask: How can you use this information to complete one square in the 15x cage? (The 15x cage must
have a 3, but the 3 cannot be in the first column because there will be a 3 in the 6x cage. Therefore,
the 3 in the 15x cage must go in the third row, second column.) Where will you place the 1 and 5?
(The 1 goes in the second row; the 5 goes in the third row.)
10. Students can continue solving by considering the 10x cage. Ask: What is the only combination that
can complete this cage? (1,2,5) Where will you place the 5? (fifth row, second column) Why?
(Because there already are 5 s in the third column and in the fourth row.) Where will you place the
1? Why? (fifth row, third column; because there already is a 1 in the fourth row.) What is the
remaining number in this cage? (2)
11. Suggest that students consider the cages in this order to finish solving Puzzle 3B: 8x, 6x in first
column, 4x, 20x, 12x, 6x in first row, 24x.
EXTEND THE LESSON
Challenge students to solve Puzzles 3C and 3D. Have students begin by identifying cages in each
puzzle that only have one possible arrangement.
Puzzle 3C: 2x (1,1,2) with the 2 in the corner; 14+ (4,5,5) with the 4 in the corner
Puzzle 3D: 3x (1,1,3) with the 3 in the corner; 100x (4,5,5) with the 4 in the corner
Then invite students to list possible combinations for each cage and to use these lists to help solve the
puzzles.
COMING UP IN NEXT WEEK S LESSON
Lesson 4: Talking KenKen
Students practice communicating their mathematical thinking coherently and clearly as they compare
KenKen solution strategies.
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PUZZLE ANSWERS
Puzzle 3A Puzzle 3B
Puzzle 3C Puzzle 3D
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