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### Ralph Waldo Emerson’s “Some Difference” Quote:

“The purpose of life is not to be happy. It is to be useful, to be honorable, to be compassionate, to have it make some difference that you have lived and lived well.”

### What Is a Sum-Difference Puzzle?

Sum-Difference Puzzles can be solved by elementary-aged children who have learned how to multiply and divide. For example, the factors of six and the number five have these relationships: 6 × 1 = 6 = 2 × 3, and 6 – 1 = 5 = 2 + 3. A Sum-Difference Puzzle asks that you put the factors, 6, 1, 2, and 3, in the appropriate boxes to make four true mathematical statements.

What do the terms factor pair, sum, and difference mean?

A factor pair of a number is two whole numbers that result in that number when multiplied together. 2 × 3 and 6 × 1 show the factor pairs of 6.

A sum results from adding two or more numbers together. The sum of 2 and 3 is 5.

A difference results from subtracting two numbers. The difference of 6 and 1 is 5.

### Links to Posts That Contain a Sum-Difference Puzzle:

(Links in green include the primitive puzzle upon which it is based. Links in pink are primitive puzzles.)

- 6 A Piece of Cake (6 is the smallest number with a Sum-Difference Puzzle.)
- 24 Think You Have a Snow Problem? Try This:
- 30 and Pieces of Pi
- 54 and How Many Squares Are in This Puzzle?
- 60 and Pay No Attention to That Man Behind the Curtain
- 84 and Level 2
- 96 and Level 1
- 120 and Level 5
- 150 and Level 3
- 180 and Level 1
- 210 and Level 4 (210 is the smallest number that is the top number in two different Sum-Difference Puzzles! Both puzzles are primitives.)
- 216 and Level 4
- A Forest of 240 Factor Trees
- 270 and Level 2
- 294 and Level 5
- 330 Christmas Factor Trees
- 336 and Level 2
- 384 and Level 6
- 480 The Very Inspiring Blogger Award
- Finding √486 and Level 6
- 504 and Level 3
- 540 and Level 4
- 546 and Level 3
- How Many Factors Do the Numbers Up to 600 Have?
- 630 Factor Trees and Level 2
- 720 Christmas Factor Trees
- Simplifying √726
- 750 and Level 4
- 756 and Level 3
- 840 is the Smallest Number that Is the Top Number in THREE Sum-Difference Puzzles (Since 840 = 4 × 210, two of those puzzles are multiples of a 210 primitive puzzle, but 840 also has a primitive puzzle of its own.)
- 864 Factor Trees
- Applying Divisibility Rules to 924
- 960 Factor Trees
- 990 Christmas Factor Trees
- Level 2 and Simplifying √1014
- STOP! Look How Cool a Number 1080 Is!
- How to Simplify √1176
- How I Knew Immediately that a Factor Pair of 1224 is . . .
- 1320 Christmas Factor Trees (1320 is the top number in two Sum-Difference Puzzles. One is primitive, one is not.)
- How Far Away Is 1344 from the Nearest Prime Number?
- 1350 Logic is at the Heart of This Puzzle
- 1386 What You Need to Know About the Multiplication Game
- 1470 Can You Find Factor Pairs That Make Sum-Difference?
- Celebrating 1500 with a Horse Race and Much More!
- Factors of 1536 Make Sum-Difference!
- Why Do Factor Pairs of 1560 Make Sum-Difference?
- Math Happens When Factors of 1620 Make Sum-Difference
- The next puzzle won’t be until 1710 and will be published when I write my 1710th post.

### An Example of Factor Pairs’ Sums and Differences:

840 has 16 factor pairs. The sums of three of its factor pairs equal the differences of three other of its factor pairs.

### An Application for Sum-Difference Puzzles:

The concept behind these puzzles will be useful in middle school when students need to recognize the difference in factors for similar-looking trinomials like x² + 5x – 6 and x² + 5x + 6.

### What Numbers Can Be in a Sum-Difference Puzzle?

If a² + b² = c², and all three numbers are counting numbers, we say that a, b, and c form a Pythagorean triple. Furthermore, we can make a Sum-Difference puzzle based on those numbers: the top number in the Sum-Difference puzzle will be ab/2, and the bottom number will be c. The top number is always divisible by 6, and at least one of the prime factors of the bottom number will be one more than a multiple of four.

If the Pythagorean triple is a primitive, meaning the only common factor for a, b, and c is one, then the Sum-Difference puzzle will be a primitive as well.

If the Pythagorean triple can be expressed as ka, kb, kc where k is a whole number greater than 1, then the top number will be k²ab/2 and the bottom number will be kc. (And I will say that the Sum-Difference puzzle is really just a puzzle with the top number ab/2 and the bottom number c in disguise!)

### Sum-Difference Puzzles’ Relationship to Pythagorean Triples:

Here’s a specific example:

Suppose we want to use the quadratic formula to solve

2x² – 5x + 3 = 0 and 2x² – 5x – 3 = 0.

Let’s combine the left sides of those two equations into one expression:

2x² -5x ± 3.

The discriminant would be (-5)² – 4(2)(±3)

= 5² ± 4(2)(3)

= 5² ± 2(4)(3)

= 4² + 3² ± 2(4)(3), ( That’s because 4² + 3² = 5², since 3-4-5 is a Pythagorean triple.)

= 4² ± 2(4)(3) + 3²

= (4 ± 3)²

= 7² and 1², two odd perfect squares, each with an odd whole number square root!

And the complete quadratic formula will yield two integer number solutions for each equation, 2x² – 5x + 3 = 0 and 2x² – 5x – 3 = 0.