# Conjectures and Counterexamples

## Learn how to make educated guesses, or conjectures, to problems, as well as how to disprove incorrect guesses with counterexamples.

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Conjectures and Counterexamples

### Conjectures and Counterexamples

A conjecture is an “educated guess” that is based on examples in a pattern. Numerous examples may make you believe a conjecture. However, no number of examples can actually prove a conjecture. It is always possible that the next example would show that the conjecture is false. A counterexample is an example that disproves a conjecture.

#### Evaluating Conjectures

1. Here’s an algebraic equation and a table of values for \begin{align*}n\end{align*} and with the result for \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 whole number value of \begin{align*}n\end{align*}.

Is this a valid, or true, conjecture?

No, this is not a valid conjecture. If Pablo were to continue the table to \begin{align*}n = 4\end{align*}, he would have seen 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 a counterexample.

2. Arthur is making figures for a graphic art project. He drew polygons and some of their diagonals.

Based on 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 drawn from any given vertex of the polygon.

Is Arthur’s conjecture correct? Can you find a counterexample to the conjecture?

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. This type of conjecture would need to be proven by induction.

#### Constructing a Counterexample

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.

#### Revisiting Examples

Your older brother made the conjecture that all guys must play sports. You could prove him wrong, or disprove his conjecture, by offering a counterexample.

Say your friend, John, is a guy and does not play sports. Just one counterexample is enough to disprove your brother's conjecture.

### Examples

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.

### Review

For questions 1-4, read the examples of reasoning in the real world. Do you think the conjectures are true or can you give a counterexample?

1. For the last three days Tommy has gone for a walk in the woods near his house at the same time of day. Each time he has seen at least one deer. Tommy reasons that if he goes for a walk tomorrow at the same time, he will see deer again.
2. Maddie likes to bake with substitutions to try to make the results healthier. She might substitute applesauce for butter or oat flour for white flour. She has noticed that whenever she does, the results don't rise well unless she adds more baking powder or baking soda than the recipe indicates. Maddie determines that she should add extra baking soda or powder when making any sort of substitution in baking.
3. One evening Juan saw a chipmunk in his backyard. He decided to leave a slice of bread with peanut butter on it for the creature to eat. The next morning the bread was gone. Juan concluded that chipmunks like to eat bread with peanut butter.
4. Sarah noticed that all her friends were in geometry class and reasoned that every 10th grade student is in geometry.
5. Describe an instance when you observed someone using invalid reasoning skills.

Give a counterexample for each of the following statements.

1. If \begin{align*}n\end{align*} is an integer, then \begin{align*}n^2 > n\end{align*}.
2. All numbers that end in 1 are prime numbers.
3. All positive fractions are between 0 and 1.
4. Any three points that are coplanar are also collinear.
5. All girls like ice cream.
6. All high school students are in choir.
7. For any angle there exists a complementary angle.
8. All teenagers can drive.
9. If \begin{align*}n\end{align*} is an integer, then \begin{align*}n>0\end{align*}.
10. All equations have integer solutions.

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