2.6: Conjectures and Counterexamples
- Page ID
- 2141
Educated guesses and examples that disprove them.
A conjecture is an “educated guess” that is based on examples in a pattern. A counterexample is an example that disproves a conjecture.
Suppose you were given a mathematical pattern like \(h = \dfrac{−16}{t^2}\). What if you wanted to make an educated guess, or conjecture, about \(h\)?
Use the following information for Examples 1 and 2:
A car salesman sold 5 used cars to five different couples. He noticed that each couple was under 30 years old. The following day, he sold a new, luxury car to a couple in their 60’s. The salesman determined that only younger couples by used cars.
Example \(\PageIndex{1}\)
Is the salesman’s conjecture logical? Why or why not?
Solution
It is logical based on his experiences, but is not true.
Example \(\PageIndex{2}\)
Can you think of a counterexample?
Solution
A counterexample would be a couple that is 30 years old or older buying a used car.
Example \(\PageIndex{3}\)
Here’s an algebraic equation and a table of values for \(n\) and \(t\).
\(t=(n−1)(n−2)(n−3)\)
\(n\) | \((n−1)(n−2)(n−3)\) | \(t\) |
---|---|---|
1 | \((0)(−1)(−2)\) | 0 |
2 | \((1)(0)(−1)\) | 0 |
3 | \((2)(1)(0)\) | 0 |
Solution
After looking at the table, Pablo makes this conjecture:
The value of \((n−1)(n−2)(n−3)\) is 0 for any number n.
Is this a true conjecture?
This is not a valid conjecture. If Pablo were to continue the table to n=4, he would have see that \((n−1)(n−2)(n−3)=(4−1)(4−2)(4−3)=(3)(2)(1)=6\)
In this example \(n=4\) is the counterexample.
Example \(\PageIndex{4}\)
Arthur is making figures for an art project. He drew polygons and some of their diagonals.

From these examples, Arthur made this conjecture:
If a convex polygon has \(n\) sides, then there are \(n−2\) triangles formed when diagonals are drawn from any vertex of the polygon.
Is Arthur’s conjecture correct? Or, can you find a counterexample?
Solution
The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw \(n−2\) triangles if the polygon has n sides.
Notice that we have not proved Arthur’s conjecture, but only found several examples that hold true. So, at this point, we say that the conjecture is true.
Example \(\PageIndex{5}\)
Give a counterexample to this statement: Every prime number is an odd number.
Solution
The only counterexample is the number 2: an even number (not odd) that is prime.
Review
Give a counterexample for each of the following statements.
- If \(n\) is a whole number, then \(n^2 >n\).
- All numbers that end in 1 are prime numbers.
- All positive fractions are between 0 and 1.
- Any three points that are coplanar are also collinear.
- All girls like ice cream.
- All high school students are in choir.
- For any angle there exists a complementary angle.
- All teenagers can drive.
- If \(n\) is an integer, then \(n>0\).
- All equations have integer solutions.
Review (Answers)
To see the Review answers, open this PDF file and look for section 2.5.
Resources
Vocabulary
Term | Definition |
---|---|
conjecture | A conjecture is an educated guess that is based on examples in a pattern. |
counterexample | A counterexample is an example that disproves a conjecture. |
fraction | A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number. |
Additional Resources
Interactive Element
Video: Inductive Reasoning
Activities: Conjectures and Counterexamples Discussion Questions
Study Aids: Types of Reasoning Study Guide
Practice: Conjectures and Counterexamples
Real World: Conjectures And Counterexamples