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Math Games and Puzzles

(Level III Math Teacher Resource)

Draft

(NSSAL)

C. David Pilmer

©2011

(Last Updated: December 2013)

This resource is the intellectual property of the Adult Education Division of the Nova Scotia

Department of Labour and Advanced Education.

The following are permitted to use and reproduce this resource for classroom purposes.

Nova Scotia instructors delivering the Nova Scotia Adult Learning Program

Canadian public school teachers delivering public school curriculum

Canadian nonprofit tuition-free adult basic education programs

The following are not permitted to use or reproduce this resource without the written

authorization of the Adult Education Division of the Nova Scotia Department of Labour and

Advanced Education.

Upgrading programs at post-secondary institutions (exception: NSCC)

Core programs at post-secondary institutions (exception: NSCC)

Public or private schools outside of Canada

Basic adult education programs outside of Canada

Individuals, not including teachers or instructors, are permitted to use this resource for their own

learning. They are not permitted to make multiple copies of the resource for distribution. Nor

are they permitted to use this resource under the direction of a teacher or instructor at a learning

institution.

NSSAL i Draft

©2011 C. D. Pilmer

Table of Contents

Introduction (for Instructors) ……………………………………………………………… ii

3 by 3 KenKen Puzzles (A to D) …………………………………………………………. 1

4 by 4 KenKen Puzzles (A and B)………………………………………………………… 5

5 by 5 KenKen Puzzles ……………………………………………………………………. 8

KenKen Puzzles: Signed Numbers ……………………………………………………… 9

Magic Squares ……………………………………………………………………………. 12

Addition Pyramids: Whole Numbers …………………………………………………….. 13

Addition Pyramids: Decimal Numbers …………………………………………………… 15

Addition Pyramids: Signed Numbers …………………………………………………….. 16

Row Factors and Column Factors ………………………………………………………… 17

Whole Number Cross Word Puzzles (A to D) …………………………………………… 18

Signed Numbers Cross Word Puzzles (A and B) ………………………………………… 22

RAD Puzzles: Whole Numbers …………………………………………………………... 24

RAD Puzzles: Signed Numbers ………………………………………………………….. 27

Connect Four Whole Number Addition Game (A and B) ……………………………….. 30

Connect Four Whole Number Subtraction Game (A and B) …………………………….. 32

Connect Four Whole Number Multiplication Game (A to D) …………………………… 34

Connect Four Whole Number Division Game …………………………………………… 38

Divisibility or Prime Connect Four Game ……………………………………………….. 39

Connect Four Fraction Decimal Equivalency Game …………………………………….. 40

Connect Four Adding Decimal Numbers Game (A and B) ……………………………… 41

Connect Four Subtracting Decimal Numbers Game (A and B) …………………………. 43

Connect Four Fraction Percent Equivalency Game ……………………………………… 45

Connect Four Percentage Game …………………………………………………………. 46

Connect Four Adding Signed Numbers Game …………………………………………… 47

Connect Four Subtracting Signed Numbers Game ………………………………………. 48

Connect Four Multiplying Signed Numbers Game ……………………………………… 49

Connect Four Dividing Signed Numbers Game …………………………………………. 50

Connect Four Squaring and Cubing of Signed Numbers Game …………………………. 51

Connect Four Time Ahead Game (A and B) …………………………………………….. 52

Connector ………………………………………………………………………………… 54

Fraction Fury Puzzles (A and B) …………………………………………………………. 55

Math Logic Puzzles ………………………………………………………………………. 59

Answers ………………………………………………………………………………….. 60

NSSAL ii Draft

©2011 C. D. Pilmer

Introduction (for Instructors)

One of the ongoing concerns for teachers, instructors, and professors, who teach secondary and

post-secondary mathematics courses, is poor arithmetic skills (and related estimation skills)

displayed by some learners. These educators are attempting to teach higher level mathematical

concepts to their learners but, in some cases, these efforts are impeded when learners have poor

arithmetic and estimation skills. These learners waste valuable time and effort and/or fail to

understand underlying concepts because of deficiencies in this area. For example, how can a

learner factor a trinomial by inspection, if one does not know their whole number math facts?

Similarly, how can a learner simplify a rational expression, if they have difficulty working with

fractions?

In the past, the approach to fostering strong arithmetic skills was to have learners complete a

variety of "drill-and-kill" questions, usually in a timed situation. Examples of such questions are

shown below.

Complete the following questions within the next ten minutes.

(a) 39 85 (b) 3 2

6 18 3 (c) Convert

72

16 to a decimal.

(d) 29458 (e) 45.8 + 682.3 (f) 4 1

2 15 6

(g) 563 28 6137 (h)

26 15 12

31 4 7

(i)

3 26 5 7 8 1

Learners who were unable to correctly answer 80% of the questions during the allotted time

would often be expected to return at lunch time or after school to make the necessary corrections.

For those adults schooled during the 1960s and 1970s, this was a common practice.

Although this practice did result in stronger arithmetic skills and in some cases stronger

estimation skills, they were two shortcomings associated with it.

1. The focus was primarily on the mastery of specific algorithms. Instead of thinking flexibly

about mathematics, learners were largely expected to follow the same rules to answer

questions. Therefore this feeds the misconception that mathematics is a rule-driven non-

creative discipline.

2. A timed test, with lunch hour or afterschool corrections, was not fun anyone. Who enjoys

math when the only reward is avoiding a correction session (i.e. detention). Also, learner

perception was that the only thing valued by the math teacher was the right answer; all the

work that preceded it was moot if the learner made a careless mistake in their last step. The

"all-that-counts-is-the-final-answer" misconception is fostered by this practice.

Does that mean that we never expose our learners to these types of "drill and kill" questions?

No, but we must recognize that these questions are only one tool for improving arithmetic skill

and that they must be used judiciously.

NSSAL iii Draft

©2011 C. D. Pilmer

Are there non-threatening and engaging means of improving arithmetic skills that also foster

more flexible thinking? Yes, and this can be accomplished using mathematical games and

puzzles. Hence we have created the following outcome for our Level III Math course.

Learners will be expected to develop efficient strategies, high levels of automaticity, and

flexible thinking skills as they pertain to arithmetic skills in the context of whole numbers,

decimal numbers, fractions, and signed numbers through the ongoing use of games and

puzzles.

In this accompanying Level III Math resource, instructors can find a variety of games and

puzzles that range from Level I to Level III. This being said, not all puzzles and games are

appropriate for all learners. Therefore the material in this resource should not be viewed as a

unit that a learner completes from "stem-to-stern" within an allotted time; rather, this is an

instructor resource where activities are gradually, yet regularly, distributed based on the

instructor's professional judgment.

NSSAL 1 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (A)

Insert the numbers 1, 2, and 3 into the grid such that:

no number is repeated in the same row or column, and

the numbers in the cages produce the target number using the indicated operation (e.g.

2 : find two numbers when multiplied give you 2).

(a) 2 5+ (b) 9 3+

4+ 6 3+

6

(c) 4+ 6 (d) 12 3

5+ 4+

3 3+

(e) 4+ 6 (f) 6 3+

3+ 5+

2 3 3

NSSAL 2 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (B)

Instructions: Insert the numbers 3, 4, and 5 into the grid such that:




(a) 12 8+ (b) 15 20

4 9+ 12

15 5 7+

(c) 15 4 (d) 8+ 12+

7+ 8+ 12

20 9+

(e) 12+ 11+ (f) 12 8+

15 14+ 12

5

NSSAL 3 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (C)





(a) 35 13+ (b) 42 11+

5 11+ 12+

42 30 7

(c) 30 19+ (d) 17+ 35

11+ 18+

13+ 7

(e) 42 5 (f) 17+ 42

11+ 13+ 13+

35 30

NSSAL 4 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (D)





(a) 72 16+ (b) 9 56

8 15+ 15+ 63

63 17+

(c) 24+ 56 (d) 17+ 56

72 23+ 72

16+

(e) 15+ 72 (f) 63 72

25+ 17+ 22+

56

NSSAL 5 Draft

©2011 C. D. Pilmer

4 by 4 KenKen Puzzles (A)

Insert the numbers 1, 2, 3, and 4 into the grid such that:




(a) 8 4+ 12 (b) 3 2 4

1 6+ 6+ 7+

6+ 6 4+

4 6 1 8

(c) 6 12 (d) 6+ 3 4

12 1 7+ 3+

9+ 3 24 1

2 8+ 2

NSSAL 6 Draft

©2011 C. D. Pilmer

4 by 4 KenKen Puzzles (B)

Insert the indicated numbers into the grid such that:




(a) 1, 2, 3, 4 Puzzle (b) 1, 2, 3, 4 Puzzle

3 8 6 6 5+

5+ 12 3+ 3

5+ 12 4 9+ 2

3+ 5+

(c) 2, 3, 4, 5 Puzzle (d) 2, 3, 4, 5 Puzzle

8 8+ 5+ 15 12+ 6+

20 6+

15 12 10 8 6 8+

6+ 5

NSSAL 7 Draft

©2011 C. D. Pilmer

(e) 3, 4, 5, 6 Puzzle (f) 3, 4, 5, 6 Puzzle

24 7+ 8+ 11+ 11+ 8+

11+ 30

12 30 10+ 18 10+

15 15 4

(g) 4, 5, 6, 7 Puzzle (h) 4, 5, 6, 7 Puzzle

11+ 11+ 42 28 15+

12+ 4 20 13+

30 17+ 20 35 42

11+ 4

(i) 5, 6, 7, 8 Puzzle (j) 6, 7, 8, 9 Puzzle

48 15+ 30 13+ 15+ 72

11+ 56 14+ 56

35 17+ 42

12+ 14+ 63

NSSAL 8 Draft

©2011 C. D. Pilmer

5 by 5 KenKen Puzzles

Insert the numbers the appropriate numbers into the grid such that:



8 : find the numbers when multiplied give you 8).

(a) 1, 2, 3, 4, 5 Puzzle (b) 2, 3, 4, 5, 6 Puzzle

8 20 6 18 5 6+

4+ 15 3+ 8 6 14+

5 4 9+ 30

4 2 8+ 3 6+ 24

5+ 9+ 11+ 5+

(c) 4, 5, 6, 7, 8 Puzzle (d) 5, 6, 7, 8, 9 Puzzle

32 42 15+ 30 9 40 12+ 42

4 40 48 54

11+ 19+ 13+ 63 45

35 48 9+ 20+ 6 72

7 10+ 13+

NSSAL 9 Draft

©2011 C. D. Pilmer

KenKen Puzzles: Signed Numbers

Insert the numbers the appropriate numbers into the grid such that:



-8 : find the numbers when multiplied give you -8).

(a) 1, -2, 3 Puzzle (b) -1, 2, -3 Puzzle (c) 2, -3, 4 Puzzle

-2 4+ -1+ 1+ 8 1+

3 -1+ 3 -6

-6 -2 -3 4 -1+

(d) -3, 4, -5 Puzzle (e) 3, -4, 5 Puzzle (f) -4, 5, -6 Puzzle

-20 -4+ -1+ -20 -5+ -30

15 -12 8+ 24

-12 5 1+

(g) 6, -7, 8 Puzzle (h) 7, -8, 9 Puzzle (i) -7, 8, -9 Puzzle

48 -7 -7+ 63 -72 -56

-56 14+ 16+ -25+

-1+ -56 1+

NSSAL 10 Draft

©2011 C. D. Pilmer

(j) 1, -2, 3, -4 Puzzle (k) 2, -3, 4, -5 Puzzle

-4 4+ 8 -1+ -5 8

-2 -6 -20 -12

-3+ -1+ -1+

1 -1+ 2 15

(l) 3, -4, 5, -6 Puzzle (m) 4, -5, 6, -7 Puzzle

-1+ -18 -30 -42 -1+ 6

15 -10+ -28 -20

-20 -5 -1+ 35

-1+ -12 24

(n) 5, -6, 7, -8 Puzzle (o) 6, -7, 8, -9 Puzzle

-30 48 6+ -72 -7 -42

7 -56 -1+ -1+

-1+ -1+ 5+

-40 63 48

NSSAL 11 Draft

©2011 C. D. Pilmer

(p) 1, -2, 3, -4, 5 Puzzle (q) 2, -3, 4, -5, 6 Puzzle

-6 -20 -5+ -18 -5 8

15 -1+ -12 15 6+ 3+

-3+ 5 12 5+

1 -1+ 3+ 4 -8+ 8+

-10 3 -10 -12

(r) -4, 5, -6, 7, -8 Puzzle (s) -5, 6, -7, 8, -9 Puzzle

1+ 48 -28 -42 8 -1+ 45 -40

-8 -30 -54 63

12+ -7+ -30 -72 -42

-14+ 3+ -42 9+ 8 -16+

-4 -40 -12+

NSSAL 12 Draft

©2011 C. D. Pilmer

Magic Squares

In a magic square, the numbers in each column, row, and diagonal all

add up to the same number. For example, with the magic square on the

right, the numbers in each column, row, and diagonal all add up to 30.

Complete each of the magic squares below.

(a) 6

3 5 7

(b) 3

2

7 5

(c) 7

6 4

5

(d) 4

6 1 8

(e) 4

8

5 12

(f) 10 3 8

7

(g) 5

9 2 7

(h) 4

7

10 3

(i) 12

9

8 6

7 14 9

12 10 8

11 6 13

NSSAL 13 Draft

©2011 C. D. Pilmer

Addition Pyramids: Whole Numbers

With addition pyramids, the two numbers in adjoining boxes add to give the number in the box

immediately above.

8 18 34

3 5 7 11 14 20

5 2 9 5 9 11

4 1 8 3

Insert the missing numbers in each of the following addition pyramids.

1. 2. 9 3. 13

4 6 5 10

3. 4. 14 6. 9

8 2 9 8

7. 8. 9.

9 9

7 2 10 3 6 2 4

10. 18 11. 12. 19

11

3 8 10 1 7 6

13. 14. 21 15.

7 12 9 14

3 2 4 8

NSSAL 14 Draft

©2011 C. D. Pilmer

16. 17. 18. 30

16 10

7 7

6 0 5 8 2 1 5

19. 20. 21.

13 12

7 12 8

3 9 2 7 3 2 6

22. 23. 24. 40

17 20 22

10 9 11 12 10

3 6 5

NSSAL 15 Draft

©2011 C. D. Pilmer

Addition Pyramids: Decimal Numbers

Complete the following addition pyramids. With addition pyramids, the two numbers in

adjoining boxes add to give the number in the box immediately above.

1. 2. 3. 3.8

1.2 2

0.4 2.1 1.7 0.8 0.2 1.4

4. 5. 3.4 6.

1.4 1.1 0.9 1.4

0.9 0.3 0.7 0.6

7. 3.1 8. 9. 4.3

2.3 1.8

1.3 0.4 1.4 0.8 1.1

10. 11. 12.

2.7

1.2 0.9 1.3 0.8

0.3 1.4 0.9 2.1 0.3 0.6 0.1

13. 14. 5.7 15.

3.4 2.6 3.2 2.7

1.6 1.8 1.7 1.8

0.2 0.4 0.3

16. 9.7 17. 18. 5.3

4.3 4.6

2.4 2.5 1.6 0.9

0.9 0.7 1.1 0.4

NSSAL 16 Draft

©2011 C. D. Pilmer

Addition Pyramids: Signed Numbers

Complete the following addition pyramids. With addition pyramids, the two numbers in

adjoining boxes add to give the number in the box immediately above.

1. 2. 3.

2 -2

-6 -5 2 -6 -4 -4 -3

4. 4 5. -2 6.

-3 -6 5

-1 -4 -5 4

7. 3 8. 9. -6

-2 -6 3

6 -3 -1 -2

10. 11. 12.

-2 -5 -1

3 4

-3 7 -2 -4 -8 -2 -3

13. 14. 15. 3

-6 -5

-2 -8 -4 -3 6

1 -5 3 -4

16. 17. -3 18.

-1 2 -6

2 -1 -1 2

-5 -4 -6 4

NSSAL 17 Draft

©2011 C. D. Pilmer

Row Factors and Column Factors

In each question you have been provided with a chart that is missing four numbers. These

numbers are the factors of the numbers found to the right of each row, and factors of the numbers

found at the bottom of each column. Find the missing numbers.

Example: Answer:

15 5 3 15

28 7 4 28

35 12 35 12

Questions:

(a) 10 (b) 18 (c) 20

12 2 12

8 15 6 6 8 30

(d) 27 (e) 60 (f) 28

7 8 3

21 9 24 20 4 21

(g) 18 (h) 18 (i) 40

20 45 7

36 10 15 54 8 35

(j) 24 (k) 20 (l) 14

18 42 32

27 16 30 28 28 16

(m) 54 (n) 56 (o) 35

40 6 27

72 30 21 16 15 63

NSSAL 18 Draft

©2011 C. D. Pilmer

Whole Number Crossword Puzzle (A)

A B C D E

F G

H I

J K

L M N

O P Q

R S T U

V W

Across:

A. Next even number after 384

C. 22 + 10 + 10

G. one thousand, four hundred twenty

I. 5 more than 228

J. Double 25

L. The product of 4 and 8

O. 196 + 231

Q. 143 - 87

R. 5 times 7

T. The number of minutes in 1 hour and 34

minutes

V. 42

W. A number between 10 and 20 that is

divisible by both 5 and 3

Down:

B. 810

D. 2000 + 100 + 30 + 9

E. 337 - 10

F. 890

H. 7 less than 470

K. Next number in the following sequence.

70, 74, 78, 82, ____

M. 3 sets of 9

N. increase 734 by 20

P. 11 + 5 + 3 + 9

S. Next number in the following sequence.

63, 60, 57, 54, ____

U. The number of cents in 2 quarters, 1 dime,

and 1 nickel

NSSAL 19 Draft

©2011 C. D. Pilmer

Whole Number Crossword Puzzle (B)

A B C D E

F G

H I

J K

L M N

O P Q

R S T U

V W

Across:

A. 509

C. 5 more than 81

G. 4000 + 800 + 10 + 5

I. increase 153 by 30

J. 63 - 29

L. ____7 = 4

O. The number of minutes in 6 hours and 4

minutes

Q. A number between 10 and 20 that is

divisible by 2, 3, 6, and 9

R. decrease 70 by 7

T. The product of 2 and 7

V. 5 + 10 + 2 + 30

W. The even number before 88

Down:

B. The next odd number after 51

D. six thousand, four hundred thirty-nine

E. 213 rounded to the nearest tens

F. Next number in the following sequence

394, 399, 404, 409, ____

H. 156 + 316

K. 72

M. Double 12

N. 15423

P. 6 sets of 11

S. 6 less than double 20

U. Next number in the following sequence

60, 54, 48, 42, ____

NSSAL 20 Draft

©2011 C. D. Pilmer

Whole Number Crossword Puzzle (C)

A B C D E

F G

H I

J K

L M N

O P Q

R S T U

V W

Across:

A. 708

C. Triple 6 plus 1

G. 7558 rounded to the nearest hundreds

I. increase 361 by 40

J. Number of cents in 3 quarters and 2 dimes

L. 9 times 6

O. 800 + 70 + 4

Q. 37 + 56

R. 5817

T. Double 13

V. Next number in the following sequence

50, 47, 44, 41, ____

W. ____6 = 8

Down:

B. 6 + 20 + 2 + 40

D. nine thousand, seven hundred twelve

E. Next number in the following sequence

886, 890, 894, 898, 902, ____

F. 1523

H. The next multiple of 5 that follows 130

K. 8 sets of 4

M. 82

N. Number of minutes in 3 hours and 16

minutes

P. 161 - 87

S. 39 decreased by 6

U. A number between 20 and 30 that is

divisible by 2, 4, 7, and 14

NSSAL 21 Draft

©2011 C. D. Pilmer

Whole Number Crossword Puzzle (D)

A B C D E

F G

H I

J K

L M N

O P Q

R S T U

V W

Across:

A. The next odd number after 769

C. 6 sets of 3

G. six thousand, three hundred seven

I. Next number in the following sequence

338, 344, 350, 356, ____

J. 4446

L. 25+47

O. 1 + 30 + 4 + 100 + 50

Q. 6 times 7

R. ____9 = 4

T. Number of cents in 2 quarters and 3 dimes

V. A number between 40 and 50 that is a

multiple of 3, 5, 9, and 15

W. The product of 8 and 9

Down:

B. 87 decreased by 9

D. 8000 + 600 + 20 + 9

E. 746 increased by 60

F. 50 less than 784

H. Number of minutes in 2 hours and 37

minutes

K. triple 8 plus 4

M. Next number in the following sequence

107, 104, 101, 98, ____

N. 2435

P. 92

S. Double 32

U. 119 - 37

NSSAL 22 Draft

©2011 C. D. Pilmer

Signed Number Crossword Puzzle (A)

A B C D E

F G

H I

J K

L M N

O P Q

R S T U

V W

Across:

A. -67

C. 24 + (-3)

G. 2

90

I. -20 + 650

J. (-3) + 3 9

L. 78 - 2 (-1)

O. 8 sets of -2

Q. -10 + 40 + (-20)

R. 8 - 15

T. 10 less than -1

V. 30 more than -10.

W. 3 + 5 (-2)

Down:

B. -5 (-8)

D. -90 (-20)

E. 300 - (-100)

F. Next number in the following sequence

-44, -49, -54, -59, ____

H. Decrease -46 by 2.

K. 12 less than double 5

M. 6 + (-5) (-4)

N. -22 + 3

P. 2

2 10

S. 40 2 + (-8)

U. -5 (-3) - 4 (-3)

NSSAL 23 Draft

©2011 C. D. Pilmer

Signed Number Crossword Puzzle (B)

A B C D E

F G

H I

J K

L M N

O P Q

R S T U

V W

Across:

A. -3 8

C. 15 - 18

G. Double 120 plus -5

H. 40 more than -10

I. -70 (-6)

J. 2 decreased by 7

L. Find the next number in the sequence.

16, 10, 4, -2, ____, …

O. -24 - (-3)

Q. 2

10 10

R. -18 - (-5) 3

V. 2

3 4

W. 2

2 4

Down:

B. -40 (-7)

D. 3400 + (-200)

E. 8 sets of -7

F. 5 + (-50)

K. The number between 0 and 20 that is

divisible by both -6 and -9.

M. 4 5 9

N. 56 7 1

P. -5 times -5

S. 4 9 1 2

T. How many times does -4 go into -120?

U. (-3) 4 + 8

NSSAL 24 Draft

©2011 C. D. Pilmer

RAD Puzzles: Whole Numbers

Using the numbers in the table below, correctly complete each puzzle.

(a) (b)

11 - = = + = 10 =

+ - + - + - + -

12 0

= = = = = = = = = =

= 16 = + 6 = 8 = +

= = = = = = = = = =

0

+ - + - -

= = + 1 + 3 = = +

1 2 2 2 2 3 3 3 4 0 1 1 2 2 2 2 3 3

4 4 4 6 7 8 8 10 21 4 4 5 5 5 6 6 9 24

30 32 48 60

(c) (d)

= = + 28 + = =

+ - + -

7

= = = = = = = = = =

40 - = 6 = = 36 = 4

= = = = = = = = = =

9

- - + + - - +

2 = = 1 + - 5 = = +

2 2 3 3 4 4 4 4 5 0 1 1 1 2 3 3 4 5

7 10 10 21 30 31 34 36 41 6 6 9 9 9 9 11 40 42

42 80 45 54

NSSAL 25 Draft

©2011 C. D. Pilmer

(e) (f)

= = 72 - + = = 8

- - + - - +

6 3

= = = = = = = = = =

+ = 63 = 9 = 24 =

= = = = = = = = = =

40 2

+ + + - + +

+ = = + = 20 =

1 2 3 3 3 4 5 6 6 0 1 2 2 3 4 4 6 6

7 7 7 7 8 10 20 21 49 6 6 8 8 10 12 19 23 30

60 70 36 48

(g) (h)

32 + = = = = -

- + +

4 2

= = = = = = = = = =

= 28 = 2 - = 18 =

= = = = = = = = = =

1 21 36

+ + + - -

= = + - = = - 9

1 2 2 3 3 3 4 4 6 2 2 3 3 4 5 6 7 8

7 7 8 8 8 9 14 17 19 9 10 11 13 15 24 27 28 30

24 56 50 81

NSSAL 26 Draft

©2011 C. D. Pilmer

(i) (j)

- = = + = = +

+ + + + +

53 1 2

= = = = = = = = = =

6 = 30 = = 48 = - 1

= = = = = = = = = =

36

- - - + -

7 + = = = = 2

1 2 2 2 3 3 3 4 5 0 1 2 2 2 3 4 6 6

5 7 7 8 10 14 16 18 30 6 7 7 8 10 12 18 30 36

42 60 49 85

(k) (l)

- = = 5 3 = =

+ + + - +

7 10

= = = = = = = = = =

- = 64 = 4 = 32 =

= = = = = = = = = =

48

+ + - - +

7 = = = 8 =

1 1 2 6 8 8 8 8 8 1 2 2 2 3 3 4 4 6

9 10 11 12 16 22 40 42 72 6 6 8 12 13 16 24 36 61

84 88 64 72

NSSAL 27 Draft

©2011 C. D. Pilmer

RAD Puzzles: Signed Numbers

Using the numbers in the table below, correctly complete each puzzle.

(a) (b)

+ = = - 8 = =

- - + - - +

13 25

= = = = = = = = = =

= -8 = + 1 = -6 = + -2

= = = = = = = = = =

20 -1

- + + + + - +

+ = = + = = -1

-9 -7 -6 -6 -5 -4 -4 -4 -3 -8 -8 -7 -6 -4 -4 -3 -2 -2

-2 -2 -1 2 6 8 9 10 11 -2 -1 1 2 2 3 4 14 21

19 24 42 48

(c) (d)

+ = = = = +

+ + -

22 52

= = = = = = = = = =

= -4 = -6 3 = -9 =

= = = = = = = = = =

28

+ - - - +

2 + = = - + = 16 = -4

-9 -9 -8 -7 -6 -6 -2 -1 1 -8 -5 -4 -3 -3 -1 -1 2 3

2 3 4 8 9 11 15 24 30 5 7 15 16 19 20 25 33 45

36 44 72 76

NSSAL 28 Draft

©2011 C. D. Pilmer

(e) (f)

+ = = = =

+ - -

-1 -3

= = = = = = = = = =

- = 25 = 3 28 = -7 =

= = = = = = = = = =

15 21

- + + + +

-8 + = = + 14 = = +

-6 -5 -5 -4 -3 -3 -3 -1 -1 -9 -7 -5 -5 -4 -2 -1 2 3

1 2 5 6 9 10 15 24 50 3 3 4 4 7 7 9 13 14

60 75 21 63

(g) (h)

= = 10 -4 + = =

- + - + - +

9

= = = = = = = = = =

-6 = 48 = = 12 = - -1

= = = = = = = = = =

64 0 8

+ + - + + + + -

= = = =

-8 -8 -7 -6 -5 -3 -2 -1 2 -9 -6 -5 -4 -3 -3 -3 -2 -2

4 4 5 8 10 12 22 25 33 0 1 2 2 2 3 6 11 14

40 50 30 36

NSSAL 29 Draft

©2011 C. D. Pilmer

(i) (j)

+ = = + 1 = = - 16

- + - -

-9 -5

= = = = = = = = = =

-2 = 18 = 40 = 8 =

= = = = = = = = = =

+ + + + - + -

= = -3 -5 = =

-9 -6 -6 -6 -5 -4 -4 -3 -2 -8 -8 -7 -7 -5 -4 -2 -2 -1

-2 -1 0 1 1 2 2 5 9 0 2 3 4 5 5 7 12 15

16 36 20 35

(k) (l)

= = 3 = = +

+ - - + - -

= = = = = = = = = =

+ = 14 = -7 45 = -5 =

= = = = = = = = = =

-5 3 7

+ + + -

-1 = = + = = 2

-9 -8 -8 -4 -3 -3 -2 -2 -2 -9 -6 -6 -4 -3 -2 -1 2 5

-2 2 4 5 6 10 10 13 16 6 6 9 10 12 14 15 19 20

18 28 30 43

NSSAL 30 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Addition Game (A)

Number of Players: Two

Objective: The winner is the first player to connect four of his/her pieces horizontally, vertically

or diagonally.

Instructions:

1. Roll a die to see which player will go first.

2. The first player looks at the board and decides which square he/she wishes to capture. They

place two paperclips on two numbers on the Addend Strip whose sum is that desired square.

Once they have chosen the two numbers, they can capture one square with that appropriate

sum. They either mark the square with an X or place a colored counter on the square. There

may be other squares with that same sum but only one square can be captured at a time.

3. Now the second player is ready to capture a square but he/she can only move one of the

paperclips on the Addend Strip. They then mark the square with that sum using an O or a

different colored marker. If a player cannot move a single paperclip to capture a square, a

paperclip must still be moved on the addend strip in order to ensure that the game can

continue.

4. Play alternates until one player connects four squares. Remember that only one paperclip is

moved at a time. If none of the players is able to connect four, then the winner is the

individual who has captured the most squares.

Game Board:

10 12 7 13 8 11

9 11 10 12 6 13

6 14 9 11 10 9

8 11 12 7 14 11

13 10 8 6 9 10

9 7 14 10 12 8

Addend Strip:

3 4 5 6 7

NSSAL 31 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Addition Game (B)



or diagonally.

Instructions:











continue.




Game Board:

12 11 18 14 16 14

15 14 13 16 12 13

16 12 15 10 14 17

15 17 14 18 15 13

13 10 13 16 11 18

15 12 11 17 14 10

Addend Strip:

5 6 7 8 9

NSSAL 32 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Subtraction Game (A)



or diagonally.

Instructions:



place two paperclips on two numbers; one from Value 1 and one from Value 2. Once they

have chosen the two numbers, they can capture one square with that appropriate difference

(i.e. Value 1 subtract Value 2). They either mark the square with an X or place a colored

counter on the square. There may be other squares with that same difference but only one

square can be captured at a time.


paperclips. They then mark the square with that difference using an O or a different colored

marker. If a player cannot move a single paperclip to capture a square, a paperclip must still

be moved in order to ensure that the game can continue.




Game Board:

3 2 5 4 6 4

7 4 6 2 3 5

6 0 1 0 5 6

3 5 3 7 2 4

4 3 2 4 1 0

1 7 0 5 3 6

Value 1: Value 2:

13 12 11 10 9 6 7 8 9

NSSAL 33 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Subtraction Game (B)



or diagonally.

Instructions:















Game Board:

3 5 4 5 6 5

6 8 7 9 10 7

9 7 6 3 5 9

7 10 5 7 8 4

8 6 9 4 9 3

4 7 8 10 5 6

Value 1: Value 2:

15 14 13 12 5 6 7 8 9

NSSAL 34 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Multiplication Game (A)



or diagonally.

Instructions:



place two paperclips on two numbers on the Factor Strip whose product is that desired

square. Once they have chosen the two numbers, they can capture one square with that

appropriate product. They either mark the square with an X or place a colored counter on the

square. There may be other squares with that same product but only one square can be

captured at a time.


paperclips on the Factor Strip. They then mark the square with that product using an O or a


paperclip must still be moved on the fraction strip in order to ensure that the game can

continue.




Game Board:

6 45 27 5 45 8

10 0 36 18 20 15

36 8 12 4 0 36

2 18 45 27 6 12

20 4 15 0 10 9

27 12 3 6 36 20

Factor Strip:

0 1 2 3 4 5 9

NSSAL 35 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Multiplication Game (B)



or diagonally.

Instructions:







captured at a time.





continue.




Game Board:

18 2 30 8 12 24

9 54 12 18 10 6

24 5 8 6 54 20

10 30 18 5 24 3

24 4 20 12 2 18

12 54 9 30 5 8

Factor Strip:

1 2 3 4 5 6 9

NSSAL 36 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Multiplication Game (C)



or diagonally.

Instructions:







captured at a time.





continue.




Game Board:

14 63 6 28 15 30

42 12 30 63 14 10

8 21 54 18 54 21

35 15 8 28 42 12

18 54 14 63 6 35

10 28 42 12 21 18

Factor Strip:

2 3 4 5 6 7 9

NSSAL 37 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Multiplication Game (D)



or diagonally.

Instructions:







captured at a time.





continue.




Game Board:

42 12 16 8 24 48

6 72 45 54 15 18

56 24 21 16 56 20

14 30 10 40 6 27

54 18 36 12 42 21

15 72 27 14 35 10

Factor Strip:

2 3 4 5 6 7 8 9

NSSAL 38 Draft

©2011 C. D. Pilmer

Connect Four Whole Number Division Game



or diagonally.

Instructions:




have chosen the two numbers, they can capture one square with that appropriate quotient (i.e.

Value 1 divided by Value 2). They either mark the square with an X or place a colored

counter on the square. There may be other squares with that same quotient but only one



paperclips. They then mark the square with that quotient using an O or a different colored






Game Board:

6 24 12 15 12 2

8 3 6 30 4 15

18 12 10 9 8 12

6 8 2 24 6 9

30 4 15 12 4 3

6 18 9 2 10 18

Value 1: Value 2:

30 24 18 12 6 1 2 3

NSSAL 39 Draft

©2011 C. D. Pilmer

Divisibility or Prime Connect Four Game



or diagonally.

Instructions:



place one paperclip on the Tens strip and one paperclip on the Ones strip. They have now

generated a two digit number. That two digit number is either divisible by a single digit

whole number greater than 1 (i.e. 2, 3, 4, 5, 6, 7, 8, 9), or the number is a prime. The player

captures a single square that describes the number. For example if the two digit number is

14, it is divisible by 2 or 7 (of the choices we are given), then the player can capture either a

square with a 2 on it, or a square with a 7 on it. If the number is prime, then a square marked

P can be captured.


paperclips on either the Tens or Ones strip. They then mark the square that describes that

number using an O or a different colored marker. If a player cannot move a single paperclip

to capture a square, a paperclip must still be moved in order to ensure that the game can

continue.




Game Board:

6 4 7 2 6 3

P 9 6 8 P 2

5 3 P 5 4 9

4 8 9 7 3 2

7 2 4 6 8 P

6 P 9 3 2 5

Tens Strip: Ones Strip

1 2 3 1 2 4 5 6 8

NSSAL 40 Draft

©2011 C. D. Pilmer

Connect Four Fraction Decimal Equivalency Game



or diagonally.

Instructions:


2. The first player looks at the board and decides which square he/she wishes to capture. The

square with a specified decimal is captured by creating the equivalent fraction using the

numerator and denominator strips at the bottom of the page. One paperclip is placed on each

strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the

denominator, then they can capture one square labeled 0.75 (3

4 is equivalent to 0.75). They

either mark the square with an X or place a colored counter on the square. Only one square

can be captured at a time.


paperclips. They then mark the square with the equivalent decimal using an O or a different

colored marker. If a player cannot move a single paperclip to capture a square, a paperclip

must still be moved in order to ensure that the game can continue.




Game Board:

0.4 1 0.2 0.4 1 0.5

0.25 0.1 0.75 0.8 0.6 0.2

0.3 0.6 0.3 0.25 0.3 0.4

0.75 0.2 0.4 0.8 0.2 1

0.2 0.8 0.25 0.1 0.5 0.4

0.1 0.5 0.6 1 0.3 0.75

Numerator (Top) Strip: Denominator (Bottom) Strip:

1 2 3 4 4 5 10

NSSAL 41 Draft

©2011 C. D. Pilmer

Connect Four Adding Decimal Numbers Game (A)



or diagonally.

Instructions:











continue.




Game Board:

0.8 1.4 1.3 2 0.8 1.1

1.1 0.5 0.9 1.5 1.2 0.9

1.4 1.6 0.8 1.1 0.5 2

1.2 0.7 1.4 1.7 1.6 0.8

0.8 1.3 2 0.5 1.3 1.4

1.4 1.7 1.1 0.9 1.5 0.7

Addend Strip:

0.2 0.3 0.5 0.6 0.9 1.1

NSSAL 42 Draft

©2011 C. D. Pilmer

Connect Four Adding Decimal Numbers Game (B)



or diagonally.

Instructions:











continue.




Game Board:

3.8 3 4.8 2 1.6 3.4

2.2 2 2.6 4.8 0.8 3

2.6 4.2 5.2 3.8 4.4 5.2

0.8 3.4 4.4 3 1.6 4.2

2 2.6 3.8 2.2 3.4 2.6

4.4 1.6 3 5.2 4.8 2.2

Addend Strip:

0.2 0.6 1.4 2 2.4 2.8

NSSAL 43 Draft

©2011 C. D. Pilmer

Connect Four Subtracting Decimal Numbers Game (A)



or diagonally.

Instructions:















Game Board:

2.5 0.6 1.8 0.1 1.9 0.9

0.1 2 1.2 2.5 0.8 2.7

1.9 2.6 2.8 1.4 2.3 1.7

1.6 0.9 1.7 1.8 2 1.6

2 2.7 1.4 2.6 0.9 2.5

1.7 0.8 2.3 1.2 2.8 0.6

Value 1: Value 2:

3 2.8 2.2 1.9 1.1 0.2 0.3 0.5 1

NSSAL 44 Draft

©2011 C. D. Pilmer

Connect Four Subtracting Decimal Numbers Game (B)



or diagonally.

Instructions:















Game Board:

2.4 1 1.4 1.2 3.2 0.8

0.4 2.2 0.8 2.6 1.4 1

2 3.2 1.6 1.8 2 2.4

1.8 0.6 3 2.8 1.2 0.4

0 1.2 1.4 0.8 0.6 2.6

2.8 1.6 2.2 0 3 1.8

Value 1: Value 2:

3.6 3 2.4 2.2 1.6 0.4 0.8 1.2 1.6

NSSAL 45 Draft

©2011 C. D. Pilmer

Connect Four Fraction Percent Equivalency Game



or diagonally.

Instructions:


2. The first player looks at the board and decides which square he/she wishes to capture. The

square with a specified percent is captured by creating the equivalent fraction using the

numerator and denominator strips at the bottom of the page. One paper clip is placed on

each strip to do so. For example, if one chooses 3 on the numerator strip and 4 on the

denominator, then they can capture one square labeled 75% (3

4 is equivalent to 75%). They

either mark the square with an X or place a colored counter on the square. There may be

other squares with that same difference but only one square can be captured at a time.


paperclips. They then mark the square with the equivalent decimal using an O or a different

colored marker. If a player cannot move a single paperclip to capture a square, a paperclip

must still be moved in order to ensure that the game can continue.

4. Play alternates until one player connects four squares. Remember that only one player clip is



Game Board:

40% 10% 20% 100% 40% 50%

25% 100% 25% 80% 60% 20%

30% 60% 40% 50% 30% 40%

75% 20% 30% 25% 80% 100%

20% 80% 75% 10% 20% 40%

10% 50% 100% 60% 30% 75%

Numerator (Top) Strip: Denominator (Bottom) Strip:

1 2 3 4 4 5 10

NSSAL 46 Draft

©2011 C. D. Pilmer

Connect Four Percentage Game



or diagonally.

Instructions:



place two paperclips on two strips below; one on the "Percentage" strip and one on the "Of"

strip. Take the percentage of that number and capture the appropriate square (e.g. 20% of 40

allows one to capture an "8" square). They either mark the square with an X or place a

colored counter on the square. There may be other squares with that same value but only one



paperclips. They then mark the square with that value using an O or a different colored






Game Board:

10 16 10 12 8 20

30 8 3 24 15 10

2 5 18 4 25 30

25 20 10 6 16 8

6 4 12 2 3 5

18 24 15 20 12 4

Percentage: Of:

10% 15% 20% 25% 20 40 80 100 120

NSSAL 47 Draft

©2011 C. D. Pilmer

Connect Four Adding Signed Numbers Game



or diagonally.

Instructions:











continue.




Game Board:

-6 -14 7 -10 -2 -8

0 3 -8 -4 -6 16

-12 16 -2 -14 2 1

2 1 -10 9 -4 0

-4 7 -6 0 -12 -8

1 -14 9 3 -6 -2

Addend Strip:

-7 -5 -1 1 8

NSSAL 48 Draft

©2011 C. D. Pilmer

Connect Four Subtracting Signed Numbers Game



or diagonally.

Instructions:















Game Board:

-1 1 2 -4 -1 -7

12 -4 7 -10 9 12

-7 2 6 -2 -1 15

-1 4 -10 -7 1 7

15 7 -2 12 -4 6

-4 9 -7 -10 4 -1

Value 1: Value 2:

10 2 -3 -6 4 1 -2 -5

NSSAL 49 Draft

©2011 C. D. Pilmer

Connect Four Multiplying Signed Numbers Game



or diagonally.

Instructions:







captured at a time.




paperclip must still be moved on the factor strip in order to ensure that the game can

continue.




Game Board:

18 -30 -54 36 25 -12

9 10 4 -15 18 -54

4 -6 81 9 -12 -54

45 -15 25 -30 -27 25

-12 -27 4 45 10 18

-30 18 -15 36 -6 81

Factor Strip:

-9 -5 -2 3 6

NSSAL 50 Draft

©2011 C. D. Pilmer

Connect Four Dividing Signed Numbers Game



or diagonally.

Instructions:




have chosen the two numbers, they can capture one square with that appropriate quotient (i.e.

Value 1 divided by Value 2). They either mark the square with an X or place a colored

counter on the square. There may be other squares with that same quotient but only one



paperclips. They then mark the square with that quotient using an O or a different colored






Game Board:

-8 -12 9 -5 2 4

-9 15 -4 6 10 -3

-15 2 6 -8 9 12

-6 -3 10 -9 -4 -5

12 -4 -8 -5 15 6

9 -15 -6 4 6 -12

Value 1: Value 2:

-30 24 -18 12 -3 -2 2 6

NSSAL 51 Draft

©2011 C. D. Pilmer

Connect Four Squaring and Cubing of Signed Numbers Game



or diagonally.

Instructions:



place two paperclips on the two strips; one on the Base Strip and one on the Exponent Strip.

Once they have chosen the values, they can capture one square with that appropriate value.

For example, if the base value is -3, and the exponent is 2, then the player can capture a 9

square 23 9 . They either mark the square with an X or place a colored counter on the

square. Only one square can be captured at a time.


paperclips. They then mark the square with that value using an O or a different colored






Game Board:

4 125 25 -64 27 -1

-27 -8 4 -1 9 125

9 27 16 -27 -64 -8

-64 25 125 4 8 1

8 9 -1 27 25 4

4 -27 1 -8 16 9

Base Strip: Exponent Strip:

-4 -3 -2 -1 2 3 5 2 (square)

3 (cube)

NSSAL 52 Draft

©2011 C. D. Pilmer

Connect Four Time Ahead Game (A)



or diagonally.

Instructions:



place paper clips on the strips below; one on the Start Time Strip and one on the Minutes

Ahead Strip. If the player chose 3:30 (start time) and 45 (minutes ahead), then they could

capture a square labelled 4:15. They either mark the square with an X or place a colored

counter on the square. There may be other squares with this same time but only one square



paperclips. They then mark the square with that time using an O or a different colored






Game Board:

5:00 4:30 6:00 4:15 3:30 5:45

3:45 5:15 4:00 4:45 3:45 4:30

4:45 5:45 3:15 5:00 4:00 5:15

3:15 4:00 5:30 4:15 6:00 3:45

4:15 5:00 4:45 5:15 3:15 5:30

5:30 3:30 4:30 3:45 5:00 4:15

Start Time Strip Minutes Ahead Strip

3:00 3:30 4:00 4:30 5:00 15 30 45 60

NSSAL 53 Draft

©2011 C. D. Pilmer

Connect Four Time Ahead Game (B)



or diagonally.

Instructions:



place paper clips on the strips below; one on the Start Time Strip and one on the Minutes

Ahead Strip. If the player chose 2:45 (start time) and 30 (minutes ahead), then they could

capture a square labelled 3:15. They either mark the square with an X or place a colored

counter on the square. There may be other squares with this same time but only one square



paperclips. They then mark the square with that time using an O or a different colored






Game Board:

3:45 3:00 4:15 3:15 4:00 3:30

3:15 4:30 2:30 3:30 2:45 3:45

2:45 4:00 3:15 3:45 3:30 4:15

3:45 3:30 4:15 3:00 4:00 3:15

4:30 3:00 3:45 3:15 2:30 3:30

3:15 2:45 3:30 4:00 3:45 3:00

Start Time Strip Minutes Ahead Strip

2:15 2:30 2:45 3:00 3:15 3:30 15 30 45 60

NSSAL 54 Draft

©2011 C. D. Pilmer

Connector

In this two or three player game, your objective is to obtain as many points as possible by

connecting numbers in straight lines vertically, horizontally, or diagonally on the playing board

below. The point values are shown below. (The points are calculated at the end of the game.)

2 points for 2 in a row





How do you capture a number? It involves rolling three six-sided dice. You use the three

numbers rolled and the operations of addition, subtraction, multiplication, division, and

exponentiation to capture one desired number. For example, if you rolled the numbers 2, 5, and

6, here are some of the possible numbers you could capture.

1 because 2 5 6 1

3 because 2 6 5 3

7 because 2 6 5 7

15 because 6 2 5 15

28 because 6 5 2 28

31 because 25 6 31

The first player rolls and captures a single number that they desire. That number should be

marked with a symbol or color specific to that player. Then the next player rolls, captures, and

marks their desired number. This process continues until it becomes difficult to complete the

remaining numbers on the board. Tally the points at the end to determine the winner.

1 2 3 4 5 6 7

8 9 10 11

12 13 14 15 16 17 18

19 20 21 22

23 24 25 26 27 28 29

30 31 32 33

34 35 36 37 38 39 40

NSSAL 55 Draft

©2011 C. D. Pilmer

Fraction Fury Puzzles (A)

With the fraction fury puzzle, your mission is to use the six

indicated numbers to create the desired nine fractions. However,

you are not permitted to repeat a number in a row or column

(They can be repeated in a diagonal.). Hints are provided so that

you can figure out which fraction (proper or improper) belongs in

each of the nine squares. If we look at the completed puzzle on

the right, the hint for the first square (top left) could have been

"1 0.2 ." There is only one possible solution based on this hint

and it is 4

5. For the second square (top center), the hint could

have been "Simplifies to 2." The possibilities are 6 4 2

, , or 3 2 1

. The 4

2 can be eliminated

because we already used the 4 in the previous square (Can't repeat the same number in a row or

column.) At this point, you are unable to determine whether 6 2

or 3 1

belongs in square two.

You will have to look at the hints provided for the other squares before determining which of the

two possible answers is appropriate for square two. Please note that you must be comfortable

with fractions, decimals and percentages to complete these puzzles.

(a) 1, 2, 3, 4, 5, 6 Puzzle (b) 2, 3, 4, 5, 6, 7 Puzzle

Whole

Number 60% Equal to

2

3 1.2

9 5

14 14 3.5 - 2.75

2 - 0.75 300%

2 6

5 5 Little >

1

2

Between 1

and 2 Equal to

1

3

1

6 of 3

1 3

16 16 2.5 150%

Between 0.5

and 1 Double

7

10

4

5

6

3

1

2

1

3

5

2

4

6

2

6

1

4

5

3

NSSAL 56 Draft

©2011 C. D. Pilmer

(c) 3, 4, 5, 6, 7, 8 Puzzle (d) 4, 5, 6, 7, 8, 9 Puzzle

1 1

3 7 1.5 Little >

1

2 Little <

1

2 Little > 1

11

7

Little < 1

12 1

8

1

2 of 150% Equal to

3

4 175% Little < 2

Equal to 2

Between 1.5

and 2 Little > 1

9 1

14 14

3 1

8 3 Equal to

8

12

(e) 1, 2, 3, 5, 7, 9 Puzzle (f) 2, 4, 5, 6, 8, 9 Puzzle

1

3 of

6

10 3.5 Equal to

1

3

4 81

9 9

Between 1

and 2 Equal to 2

450%

5 32

8 8

Between 0.5

and 1

1

2 of 300% 3 sets of

3

2 Little >

1

2

Little <

1

2 2 - 0.2

2 11

5 4 Equal to

1

4 0.8 Equal to

2

3

NSSAL 57 Draft

©2011 C. D. Pilmer

(g) 1, 3, 4, 5, 8, 9 Puzzle (h) 1, 2, 5, 6, 7, 8 Puzzle

Between 1

and 1.5 Close to 0

2 5

3 6

1

2 of 0.8

Between 1

and 1.5

1 7

3 3

160% 2.25

Whole

Number Close to 1

Whole

Number Little > 1

7 5

18 18 50% of 1.2 Equal to 2

1 111

12 12

Between 0.5

and 1 Equal to 4

(i) 3, 4, 5, 6, 7, 8 Puzzle (j) 2, 5, 6, 7, 8, 9 Puzzle

1 3

4 8 Little >

1

2 Equal to 2 1.5 Equal to

1

4

Between 1

and 1.5

Between 2

and 3 Equal to

11

3 125% 250%

Between 1

and 1.5

Equal to

75%

Equal to

2

3

7 5

20 20 Little > 1

3 51

16 16 Little < 1 50% of

4

9

NSSAL 58 Draft

©2011 C. D. Pilmer

Fraction Fury Puzzles (B)

All of the puzzles on this page use the numbers 1 through 8.

1.

2 3

3 5 Double

4

7

Whole

Number <6 Equal to

1

2

11 8

21 21

21

3

Equal to

75% 2 sets of

1

4

Equal to 2 1 - 0.75 3.5 160%

3 7

18 8 Equal to

1

3

Between 1.5

and 2

Whole

Number >4

2.

1 32 3

4 4 Little >

1

2 2 - 1.6

Between 2.5

and 3

1

2 of

6

7 Equal to

1

4 Equal to

2

3

5 11 3

6 6

80% 2 5

5 6 Double

7

16 Equal to

1

3

8 sets of 1

2 120%

Between 2.7

and 3.6 1.75

3.

Equal to 4 600% 3 - 1.25

1

3 of

18

10

4 8

21 21

11 3

8 1 - 0.6

One part of

six

Whole

Number >4 Equal to

1

2 Triple

1

6

9 111

16 16

Equal to 1

2

5 91

14 14

1

4 of

1

2 Equal to 2

NSSAL 59 Draft

©2011 C. D. Pilmer

Math Logic Puzzles

For each, find the numbers represented by the symbols , , and . Hint: For each of the

puzzles, one of the equations, not necessarily the first equation, allows you to solve for a symbol

very quickly.

Puzzle 1:

- = 2

+ 1 = 6

+ = 8

Puzzle 2:

+ = 8

+ = 6

+ = 7

Puzzle 3:

+ = 8

3 = 6

+ = 10

Puzzle 4:

- = 3

+ + = 10

4 = 8

Puzzle 5:

- 2 = 7

+ + = 17

= 18

Puzzle 6:

- = 5

+ + = 9

+ =

Puzzle 7:

- 3 =

+ = 2

14 = 2

Puzzle 8:

+ + 2 =

3 = 4

- = 6

Puzzle 9:

24 =

+ + 1 = 9

+ = 3

Answers (in no particular order)

= 4, = 6, = 2 = 6, = 4, = 12 = 6, = 2, = 9

= 14, = 6, = 4 = 11, = 3, = 8 = 7, = 5, = 1

= 2, = 5, = 3 = 4, = 2, = 5 = 1, = 7, = 4

NSSAL 60 Draft

©2011 C. D. Pilmer

Answers

3 by 3 KenKen Puzzles (A) (page 1)

(a)

2 1 3 (b)

3 1 2

1 3 2

2 3 1

3 2 1

1 2 3

(c)

1 2 3 (d)

2 3 1

3 1 2

1 2 3

2 3 1

3 1 2

(e)

3 1 2 (f)

3 2 1

1 2 3

2 1 3

2 3 1

1 3 2

NSSAL 61 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (B) (page 2)

(a)

4 5 3 (b)

3 5 4

3 4 5

4 3 5

5 3 4

5 4 3

(c)

5 3 4 (d)

3 4 5

4 5 3

5 3 4

3 4 5

4 5 3

(e)

5 4 3 (f)

4 3 5

3 5 4

3 5 4

4 3 5

5 4 3

NSSAL 62 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (C) (page 3)

(a)

5 6 7 (b)

7 5 6

7 5 6

6 7 5

6 7 5

5 6 7

(c)

6 5 7 (d)

6 7 5

5 7 6

5 6 7

7 6 5

7 5 6

(e)

7 6 5 (f)

5 7 6

5 7 6

6 5 7

6 5 7

7 6 5

NSSAL 63 Draft

©2011 C. D. Pilmer

3 by 3 KenKen Puzzles (D) (page 4)

(a)

8 7 9 (b)

9 7 8

9 8 7

7 8 9

7 9 8

8 9 7

(c)

7 9 8 (d)

9 8 7

9 8 7

8 7 9

8 7 9

7 9 8

(e)

8 7 9 (f)

7 9 8

7 9 8

8 7 9

9 8 7

9 8 7

NSSAL 64 Draft

©2011 C. D. Pilmer

4 by 4 KenKen Puzzles (A) (page 5)

(a)

2 3 1 4 (b)

3 1 2 4

4 1 2 3

2 4 3 1

3 2 4 1

4 2 1 3

1 4 3 2

1 3 4 2

(c)

1 2 4 3 (d)

2 1 3 4

3 4 2 1

4 3 1 2

4 1 3 2

3 4 2 1

2 3 1 4

1 2 4 3

NSSAL 65 Draft

©2011 C. D. Pilmer

4 by 4 KenKen Puzzles (B) (pages 6 and 7)

(a) 1, 2, 3, 4 Puzzle (b) 1, 2, 3, 4 Puzzle

1 3 4 2

2 3 1 4

4 1 2 3

4 1 2 3

2 4 3 1

3 2 4 1

3 2 1 4

1 4 3 2

(c) 2, 3, 4, 5 Puzzle (d) 2, 3, 4, 5 Puzzle

4 5 2 3

5 3 4 2

2 3 5 4

3 2 5 4

5 4 3 2

2 4 3 5

3 2 4 5

4 5 2 3

NSSAL 66 Draft

©2011 C. D. Pilmer

(e) 3, 4, 5, 6 Puzzle (f) 3, 4, 5, 6 Puzzle

6 4 3 5

4 3 6 5

5 6 4 3

6 4 5 3

3 5 6 4

5 6 3 4

4 3 5 6

3 5 4 6

(g) 4, 5, 6, 7 Puzzle (h) 4, 5, 6, 7 Puzzle

4 6 5 7

7 4 6 5

7 5 4 6

5 6 7 4

5 7 6 4

4 7 5 6

6 4 7 5

6 5 4 7

NSSAL 67 Draft

©2011 C. D. Pilmer

(i) 5, 6, 7, 8 Puzzle (j) 6, 7, 8, 9 Puzzle

8 6 7 5

7 9 6 8

5 7 8 6

6 8 7 9

6 8 5 7

9 6 8 7

7 5 6 8

8 7 9 6

5 by 5 Ken Ken Puzzles (page 8)

(a) 1, 2, 3, 4, 5 Puzzle (b) 2, 3, 4, 5, 6 Puzzle

2 5 4 1 3 6 3 5 4 2

4 3 5 2 1 4 2 3 6 5

5 1 3 4 2 2 4 6 5 3

1 4 2 3 5 3 5 4 2 6

3 2 1 5 4 5 6 2 3 4

NSSAL 68 Draft

©2011 C. D. Pilmer

(c) 4, 5, 6, 7, 8 Puzzle (d) 5, 6, 7, 8, 9 Puzzle

4 7 6 8 5 9 5 8 7 6

8 4 5 7 6 6 8 9 5 7

6 5 8 4 7 8 7 6 9 5

7 6 4 5 8 5 9 7 6 8

5 8 7 6 4 7 6 5 8 9

KenKen Puzzles: Signed Numbers (pages 9 to 11)

(a) 1, -2, 3 Puzzle (b) -1, 2, -3 Puzzle (c) 2, -3, 4 Puzzle

-2 1 3 -3 -1 2 2 4 -3

1 3 -2 2 -3 -1 -3 2 4

3 -2 1 -1 2 -3 4 -3 2

(d) -3, 4, -5 Puzzle (e) 3, -4, 5 Puzzle (f) -4, 5, -6 Puzzle

-5 -3 4 3 5 -4 5 -4 -6

4 -5 -3 -4 3 5 -4 -6 5

-3 4 -5 5 -4 3 -6 5 -4

NSSAL 69 Draft

©2011 C. D. Pilmer

(g) 6, -7, 8 Puzzle (h) 7, -8, 9 Puzzle (i) -7, 8, -9 Puzzle

6 8 -7 -8 9 7 -9 -7 8

-7 6 8 7 -8 9 8 -9 -7

8 -7 6 9 7 -8 -7 8 -9

(j) 1, -2, 3, -4 Puzzle (k) 2, -3, 4, -5 Puzzle

-4 3 1 -2 -3 2 -5 4

1 -2 3 -4 -5 -3 4 2

3 -4 -2 1 4 -5 2 -3

-2 1 -4 3 2 4 -3 -5

(l) 3, -4, 5, -6 Puzzle (m) 4, -5, 6, -7 Puzzle

-4 3 -6 5 -7 -5 4 6

3 -6 5 -4 6 -7 -5 4

5 -4 3 -6 -5 4 6 -7

-6 5 -4 3 4 6 -7 -5

NSSAL 70 Draft

©2011 C. D. Pilmer

(n) 5, -6, 7, -8 Puzzle (o) 6, -7, 8, -9 Puzzle

5 -8 -6 7 8 -9 -7 6

-6 7 -8 5 6 8 -9 -7

-8 5 7 -6 -7 6 8 -9

7 -6 5 -8 -9 -7 6 8

(p) 1, -2, 3, -4, 5 Puzzle (q) 2, -3, 4, -5, 6 Puzzle

3 -4 5 1 -2 6 -3 -5 2 4

-2 5 1 -4 3 -5 4 2 6 -3

1 3 -2 5 -4 -3 2 6 4 -5

-4 1 3 -2 5 4 6 -3 -5 2

5 -2 -4 3 1 2 -5 4 -3 6

(r) -4, 5, -6, 7, -8 Puzzle (s) -5, 6, -7, 8, -9 Puzzle

5 -6 -8 -4 7 8 -7 6 -9 -5

-4 -8 5 7 -6 -9 6 -7 -5 8

7 5 -6 -8 -4 -5 8 -9 -7 6

-8 -4 7 -6 5 6 -9 -5 8 -7

-6 7 -4 5 -8 -7 -5 8 6 -9

NSSAL 71 Draft

©2011 C. D. Pilmer

Magic Squares (page 12)

(a) 8 1 6

3 5 7

4 9 2

(b) 3 8 1

2 4 6

7 0 5

(c) 7 2 9

8 6 4

3 10 5

(d) 2 9 4

7 5 3

6 1 8

(e) 9 4 11

10 8 6

5 12 7

(f) 10 3 8

5 7 9

6 11 4

(g) 5 10 3

4 6 8

9 2 7

(h) 6 11 4

5 7 9

10 3 8

(i) 12 5 10

7 9 11

8 13 6

Addition Pyramids: Whole Numbers (pages 13 and 14)

1. 10 2. 9 3. 13

4 6 5 4 3 10

3. 10 4. 14 6. 9

8 2 5 9 8 1

NSSAL 72 Draft

©2011 C. D. Pilmer

7. 21 8. 18 9. 16

9 12 9 9 7 9

7 2 10 6 3 6 2 5 4

10. 18 11. 29 12. 19

7 11 18 11 13 6

4 3 8 8 10 1 7 6 0

13. 19 14. 21 15. 24

7 12 9 12 10 14

4 3 9 2 7 5 4 6 8

16. 29 17. 26 18. 30

11 18 10 16 10 20

6 5 13 3 7 9 3 7 13

6 0 5 8 2 1 6 3 1 2 5 8

19. 25 20. 50 21. 28

13 12 27 23 15 13

8 5 7 12 15 8 10 5 8

5 3 2 5 3 9 6 2 7 3 2 6

22. 36 23. 39 24. 40

17 19 19 20 18 22

7 10 9 11 8 12 6 12 10

3 4 6 3 5 6 2 10 1 5 7 3

NSSAL 73 Draft

©2011 C. D. Pilmer

Addition Pyramids: Decimal Numbers (page 15)

1. 6.3 2. 2.2 3. 3.8

2.5 3.8 1.2 1 1.8 2

0.4 2.1 1.7 0.4 0.8 0.2 0.4 1.4 0.6

4. 2.2 5. 3.4 6. 2.3

1.4 0.8 2.3 1.1 0.9 1.4

0.9 0.5 0.3 1.9 0.4 0.7 0.3 0.6 0.8

7. 3.1 8. 4.5 9. 4.3

1.4 1.7 2.2 2.3 1.8 2.5

0.1 1.3 0.4 1.4 0.8 1.5 0.7 1.1 1.4

10. 9.3 11. 5.1 12. 3.7

4 5.3 2.7 2.4 2.1 1.6

1.7 2.3 3 1.2 1.5 0.9 1.3 0.8 0.8

0.3 1.4 0.9 2.1 0 1.2 0.3 0.6 0.6 0.7 0.1 0.7

13. 6 14. 5.7 15. 6.2

3.4 2.6 2.5 3.2 3.5 2.7

1.6 1.8 0.8 1.1 1.4 1.8 1.7 1.8 0.9

0 1.6 0.2 0.6 0.7 0.4 1 0.8 0.5 1.2 0.6 0.3

16. 9.7 17. 8.9 18. 5.3

4.3 5.4 4.3 4.6 2.5 2.8

1.9 2.4 3 1.8 2.5 2.1 1.6 0.9 1.9

0.4 1.5 0.9 2.1 0.7 1.1 1.4 0.7 1.2 0.4 0.5 1.4

NSSAL 74 Draft

©2011 C. D. Pilmer

Addition Pyramids: Signed Numbers (page 16)

1. -14 2. 2 3. -9

-11 -3 0 2 -2 -7

-6 -5 2 -6 6 -4 2 -4 -3

4. 4 5. -2 6. -1

7 -3 -9 7 -6 5

8 -1 -2 -4 -5 12 -10 4 1

7. 3 8. -10 9. -6

-2 5 -6 -4 3 -9

6 -8 13 -3 -3 -1 5 -2 -7

10. 8 11. -7 12. -2

9 -1 -2 -5 -1 -1

4 5 -6 3 -5 0 -5 4 -5

-3 7 -2 -4 0 3 -8 8 -11 6 -2 -3

13. -14 14. -13 15. 3

-4 -10 -7 -6 -5 8

-2 -2 -8 -4 -3 -3 6 -11 19

-3 1 -3 -5 -7 3 -6 3 13 -7 -4 23

16. -8 17. -3 18. -5

-1 -7 -5 2 -6 1

-3 2 -9 -4 -1 3 -5 -1 2

-10 7 -5 -4 2 -6 5 -2 4 -9 8 -6

NSSAL 75 Draft

©2011 C. D. Pilmer

Factor Rows and Factor Columns (page 17)

(a) 2 5 10 (b) 6 3 18 (c) 4 5 20

4 3 12 1 2 2 2 6 12

8 15 6 6 8 30

(d) 3 9 27 (e) 6 10 60 (f) 4 7 28

7 1 7 4 2 8 1 3 3

21 9 24 20 4 21

(g) 9 2 18 (h) 3 6 18 (i) 8 5 40

4 5 20 5 9 45 1 7 7

36 10 15 54 8 35

(j) 3 8 24 (k) 5 4 20 (l) 7 2 14

9 2 18 6 7 42 4 8 32

27 16 30 28 28 16

(m) 9 6 54 (n) 7 8 56 (o) 5 7 35

8 5 40 3 2 6 3 9 27

72 30 21 16 15 63

NSSAL 76 Draft

©2011 C. D. Pilmer

Whole Number Crossword Puzzle (A) (page 18)

3 8 6 4 2 3

0 7 1 4 2 0

4 2 3 3 7

6 5 0 9 8

3 2 2 7 6

4 2 7 5 6

3 5 8 9 4 6

1 6 1 5

Whole Number Crossword Puzzle (B) (page 19)

4 5 0 8 6 2

3 4 4 8 1 5

4 1 8 3 0

7 3 4 9 4

2 8 2 5 9

3 6 4 1 8

6 3 6 1 4 3

4 7 8 6

NSSAL 77 Draft

©2011 C. D. Pilmer

Whole Number Crossword Puzzle (C) (page 20)

5 6 0 1 9 9

8 3 7 6 0 0

1 4 0 1 6

3 9 5 2 3

5 4 6 1 2

8 7 4 9 3

8 3 4 2 6 2

3 8 4 8

Whole Number Crossword Puzzle (D) (page 21)

7 7 1 1 8 8

8 7 6 3 0 7

1 3 6 2 6

5 7 4 9 2

7 2 9 8 8

1 8 5 4 2

3 6 1 8 0 8

4 5 7 2

NSSAL 78 Draft

©2011 C. D. Pilmer

Signed Number Crossword Puzzle (A) (page 22)

- 4 2 2 1 4

0 - 8 1 0 0

- 6 3 0 0

4 2 4 0 -

8 0 2 - 2

- 1 6 1 0

- 7 4 - 9 2

2 0 - 7

Signed Number Crossword Puzzle (B) (page 23 )

- 2 4 - 3 -

8 - 2 3 5

3 0 4 2 0 6

- 5 0 1

- 8 - - 8

- 2 1 9 0

- 3 5 3 -

4 9 0 1 4

NSSAL 79 Draft

©2011 C. D. Pilmer

RAD Puzzles: Whole Numbers (pages 24 to 26)

(a) (b)

11 - 3 = 8 = 6 + 2 60 6 = 10 = 2 5

+ - + - + - + -

21 1 2 4 3 12 0 2 1 0

= = = = = = = = = =

32 2 = 16 = 10 + 6 48 6 = 8 = 3 + 5

= = = = = = = = = =

4 0 4 30 7 24 2 3 4 9

+ - + - -

8 2 = 4 = 3 + 1 2 + 3 = 5 = 1 + 4

(c) (d)

10 3 = 30 = 2 + 28 42 + 3 = 45 = 9 5

+ - + -

4 31 5 21 4 7 2 9 0 1

= = = = = = = = = =

40 - 34 = 6 = 42 7 6 6 = 36 = 9 4

= = = = = = = = = =

80 36 10 41 4 54 11 40 9 1

- - + + - - +

2 2 = 4 = 1 + 3 9 - 5 = 4 = 1 + 3

NSSAL 80 Draft

©2011 C. D. Pilmer

(e) (f)

10 7 = 70 = 72 - 2 36 + 12 = 48 = 6 8

- - + - - +

6 4 7 8 5 6 8 2 3 0

= = = = = = = = = =

60 + 3 = 63 = 9 7 6 4 = 24 = 3 8

= = = = = = = = = =

40 2 3 6 49 6 23 4 30 6

+ + + - + +

20 + 1 = 21 = 3 7 1 + 19 = 20 = 10 2

(g) (h)

32 + 24 = 56 = 7 8 24 8 = 3 = 30 - 27

- + +

8 17 2 2 4 4 2 6 15 3

= = = = = = = = = =

4 7 = 28 = 14 2 28 - 10 = 18 = 2 9

= = = = = = = = = =

1 21 19 6 3 4 50 36 13 81

+ + + - -

3 3 = 9 = 8 + 1 7 - 5 = 2 = 11 - 9

NSSAL 81 Draft

©2011 C. D. Pilmer

(i) (j)

18 - 4 = 14 = 7 2 6 + 2 = 8 = 7 + 1

+ + + + +

3 1 16 53 1 2 2 6 7 0

= = = = = = = = = =

6 5 = 30 = 60 2 12 4 = 48 = 49 - 1

= = = = = = = = = =

42 8 3 30 7 36 10 30 85 2

- - - + -

7 + 3 = 10 = 2 5 3 6 = 18 = 36 2

(k) (l)

9 - 1 = 8 = 40 5 3 12 = 36 = 6 6

+ + + - +

8 7 8 10 11 61 6 4 3 10

= = = = = = = = = =

72 - 8 = 64 = 4 16 64 2 = 32 = 2 16

= = = = = = = = = =

12 1 22 88 8 72 2 4 48 13

+ + - - +

6 7 = 42 = 84 2 8 1 = 8 = 24 3

NSSAL 82 Draft

©2011 C. D. Pilmer

RAD Puzzles: Signed Numbers (pages 27 to 29)

(a) (b)

-6 + 8 = 2 = 10 - 8 3 14 = 42 = 21 2

- - + - - +

-4 11 -4 19 -7 -2 13 -7 25 -4

= = = = = = = = = =

24 -3 = -8 = -9 + 1 -6 1 = -6 = -4 + -2

= = = = = = = = = =

20 -1 -2 9 6 48 -3 -8 -2 -1

- + + + + - +

-4 + -2 = -6 = -1 + -5 -8 4 = -2 = 2 -1

(c) (d)

-8 + 44 = 36 = -6 -6 15 3 = 45 = 20 + 25

+ + -

4 22 -9 30 1 5 -1 -5 52 33

= = = = = = = = = =

-2 2 = -4 = 24 -6 3 -3 = -9 = 72 -8

= = = = = = = = = =

-1 11 28 3 9 -1 16 7 76 2

+ - - - +

2 + -9 = -7 = 8 - 15 -3 + 19 = 16 = -4 -4

NSSAL 83 Draft

©2011 C. D. Pilmer

(e) (f)

-6 + 1 = -5 = 15 -3 7 9 = 63 = 21 3

+ - -

-4 -1 -5 60 -1 4 13 -9 14 -3

= = = = = = = = = =

24 - -1 = 25 = 75 3 28 -4 = -7 = 7 -1

= = = = = = = = = =

-3 9 50 15 6 2 3 -5 21 4

- + + + +

-8 + 10 = 2 = 5 + -3 14 -7 = -2 = 3 + -5

(g) (h)

2 25 = 50 = 5 10 -4 + 0 = -4 = 2 -2

- + - + - +

-3 33 -2 -7 -6 -9 -3 -3 9 -2

= = = = = = = = = =

-6 -8 = 48 = 12 4 36 3 = 12 = 11 - -1

= = = = = = = = = =

-1 64 8 22 0 6 8 -2 14 1

+ + - + + + + -

-5 -8 = 40 = 10 4 30 -5 = -6 = -3 2

NSSAL 84 Draft

©2011 C. D. Pilmer

(i) (j)

-1 + -2 = -3 = -4 + 1 -8 -2 = 4 = 20 - 16

- + - -

9 0 -6 -9 1 -5 -7 12 -5 -8

= = = = = = = = = =

-9 -2 = 18 = 36 2 40 5 = 8 = -4 -2

= = = = = = = = = =

-5 1 16 -6 5 5 0 15 3 2

+ + + + - + -

-4 -2 = 2 = -6 -3 35 -5 = -7 = 7 -1

(k) (l)

-3 -2 = 6 = 18 3 2 10 = 20 = 14 + 6

+ - - + - -

13 -2 -8 -9 10 43 19 -4 9 -6

= = = = = = = = = =

10 + 4 = 14 = -2 -7 45 -9 = -5 = 5 -1

= = = = = = = = = =

-5 5 28 16 -3 3 -6 7 30 -2

+ + + -

-2 -1 = 2 = -8 -4 15 + -3 = 12 = 6 2

NSSAL 85 Draft

©2011 C. D. Pilmer

Fraction Fury Puzzles (A) (pages 55 to 57)

(a) 1, 2, 3, 4, 5, 6 Puzzle (b) 2, 3, 4, 5, 6, 7 Puzzle

2

1

3

5

4

6

6

5

2

7

3

4

5

4

6

2

1

3

4

7

5

3

2

6

3

6

1

4

5

2

3

2

4

6

7

5

(c) 3, 4, 5, 6, 7, 8 Puzzle (d) 4, 5, 6, 7, 8, 9 Puzzle

7

3

6

4

5

8

4

9

6

5

8

7

5

6

7

8

3

4

6

8

7

4

9

5

8

4

5

3

7

6

5

7

9

8

4

6

(e) 1, 2, 3, 5, 7, 9 Puzzle (f) 2, 4, 5, 6, 8, 9 Puzzle

1

5

7

2

3

9

5

9

8

6

4

2

9

2

3

1

5

7

6

4

9

2

5

8

3

7

9

5

1

2

2

8

4

5

6

9

NSSAL 86 Draft

©2011 C. D. Pilmer

(g) 1, 3, 4, 5, 8, 9 Puzzle (h) 1, 2, 5, 6, 7, 8 Puzzle

4

3

1

8

5

9

2

5

8

6

1

7

8

5

9

4

3

1

7

8

2

1

6

5

1

9

3

5

8

4

1

6

5

7

8

2

(i) 3, 4, 5, 6, 7, 8 Puzzle (j) 2, 5, 6, 7, 8, 9 Puzzle

5

8

4

7

6

3

9

6

2

8

7

5

7

3

8

6

5

4

5

2

9

7

6

8

4

6

3

5

8

7

7

8

5

6

2

9

Fraction Fury Puzzles (B) (page 58)

1. 2

5

8

7

4

1

3

6

1

7

5

3

6

8

2

4

6

3

1

4

7

2

8

5

4

8

2

6

5

3

7

1

NSSAL 87 Draft

©2011 C. D. Pilmer

2. 6

1

4

7

2

5

8

3

3

7

2

8

4

6

5

1

4

5

1

3

7

8

2

6

8

2

6

5

3

1

7

4

3. 8

2

6

1

7

4

3

5

4

7

3

8

2

5

1

6

5

1

2

4

3

6

7

8

3

6

5

7

1

8

4

2

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