### Conjectures and Counterexamples

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* = \begin{align*}-16/t^2\end{align*}. What if you wanted to make an educated guess, or conjecture, about *h*?

### Examples

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 1

Is the salesman’s conjecture logical? Why or why not?

It is logical based on his experiences, but is not true.

#### Example 2

Can you think of a counterexample?

A counterexample would be a couple that is 30 years old or older buying a used car.

#### Example 3

Here’s an algebraic equation and a table of values for \begin{align*}n\end{align*} and \begin{align*}t\end{align*}.

\begin{align*}t=(n-1)(n-2)(n-3)\end{align*}

\begin{align*}n\end{align*} | \begin{align*}(n-1)(n-2)(n-3)\end{align*} | \begin{align*}t\end{align*} |
---|---|---|

1 | \begin{align*}(0)(-1)(-2)\end{align*} | 0 |

2 | \begin{align*}(1)(0)(-1)\end{align*} | 0 |

3 | \begin{align*}(2)(1)(0)\end{align*} | 0 |

After looking at the table, Pablo makes this conjecture:

The value of \begin{align*}(n-1)(n-2)(n-3)\end{align*} is 0 for any number \begin{align*}n\end{align*}.

Is this a true conjecture?

This is not a valid conjecture. If Pablo were to continue the table to \begin{align*}n = 4\end{align*}, he would have see that \begin{align*}(n-1)(n-2)(n-3)=(4-1)(4-2)(4-3)=(3)(2)(1)=6\end{align*}

In this example \begin{align*}n = 4\end{align*} is the counterexample.

#### Example 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 \begin{align*}n\end{align*} sides, then there are \begin{align*}n - 2\end{align*} triangles formed when diagonals are drawn from any vertex of the polygon.

Is Arthur’s conjecture correct? Or, can you find a counterexample?

The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw \begin{align*}n - 2\end{align*} triangles if the polygon has \begin{align*}n\end{align*} 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 5

Give a counterexample to this statement: Every prime number is an odd number.

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 \begin{align*}n\end{align*} is a whole number, then \begin{align*}n^2 > n\end{align*}.
- 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 \begin{align*}n\end{align*} is an integer, then \begin{align*}n>0\end{align*}.
- All equations have integer solutions.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.5.