Sums of Odd Numbers

Add the first few odd numbers: $1$, $1 + 3$, $1 + 3 + 5$, $1 + 3 + 5 + 7$. Make a conjecture and use it to predict the sum of the first $10$ odd numbers.

Observe the cases: $1 = 1$, $1 + 3 = 4$, $1 + 3 + 5 = 9$, $1 + 3 + 5 + 7 = 16$ = Sums: $1, 4, 9, 16$

Look for a pattern: $1 = 1^2$, $4 = 2^2$, $9 = 3^2$, $16 = 4^2$ = Every sum is a square number

Make a conjecture: With $n$ odd numbers the sum seems to be $n \times n$ = Sum of the first $n$ odd numbers $= n^2$

Test it with a new case: $1 + 3 + 5 + 7 + 9 = 25 = 5^2$ = The conjecture survives the test

Use it to predict: $n = 10$: $10^2 = 100$ = $100$

Answer: Conjecture: the sum of the first $n$ odd numbers is $n^2$. The sum of the first $10$ odd numbers should be $100$ (and this conjecture has in fact been proven).

Breaking a Conjecture with Zero and Negatives

Leo tests the claim "doubling a number always makes it bigger" with $3$, $10$ and $25$. Is the claim true?

Check Leo's cases: $3 \to 6$, $10 \to 20$, $25 \to 50$ = All three got bigger

Try a different kind of number: zero: $2 \times 0 = 0$ = $0$ is not bigger than $0$

Try a negative number: $2 \times (-5) = -10$, and $-10 < -5$ = The double is smaller

Decide: One counterexample is enough = The claim is false

Answer: The claim is false: $0$ and $-5$ are counterexamples. A corrected conjecture is "doubling a positive number makes it bigger".

Diagonals of Polygons

A triangle has $0$ diagonals, a quadrilateral $2$, a pentagon $5$ and a hexagon $9$. Predict the number of diagonals of an octagon.

List the cases: $3$ sides: $0$, $4$ sides: $2$, $5$ sides: $5$, $6$ sides: $9$ = $0, 2, 5, 9$

Find the differences: $2 - 0 = 2$, $5 - 2 = 3$, $9 - 5 = 4$ = The differences grow by $1$

Continue the pattern: Heptagon: $9 + 5 = 14$, octagon: $14 + 6 = 20$ = Prediction: $20$

Test with a formula that fits all cases: $\frac{n(n - 3)}{2}$: $\frac{6 \times 3}{2} = 9$, $\frac{8 \times 5}{2} = 20$ = Both methods give $20$

Answer: An octagon should have $20$ diagonals. Counting them in a drawing confirms it.

Forty Cases and Still Wrong

Test the conjecture "$n^2 + n + 41$ is always a prime number" for $n = 0, 1, 2, 3$ and then for $n = 40$.

Test small values: $n = 0$: $41$, $n = 1$: $43$, $n = 2$: $47$, $n = 3$: $53$ = All prime

Notice how far it goes: It keeps giving primes up to $n = 39$ = $40$ cases in a row work

Test $n = 40$: $40^2 + 40 + 41 = 1600 + 40 + 41$ = $1681$

Check if $1681$ is prime: $1681 = 41 \times 41$ = Not prime: a counterexample

Answer: The conjecture is false. $n = 40$ is a counterexample, even though the first $40$ cases all worked.

Inductive and Deductive Side by Side

Mia notices: $3 + 5 = 8$, $7 + 9 = 16$, $11 + 1 = 12$. What can she conclude about the sum of two odd numbers, and how can she be sure?

Reason inductively: $8$, $16$ and $12$ are all even = Conjecture: odd $+$ odd is even

See the limit: Three cases cannot check every pair of odd numbers = Only probably true so far

Reason deductively: Odd numbers are $2a + 1$ and $2b + 1$: $(2a + 1) + (2b + 1) = 2(a + b + 1)$ = Always a multiple of $2$

Compare: The examples gave the idea, the algebra gave the proof = Now it is certain

Answer: Inductive reasoning gives the conjecture "odd $+$ odd is even". Deductive reasoning with $2(a + b + 1)$ proves it for all odd numbers.

Mistake: Thinking a conjecture is proven because it worked in many cases

Why: Examples can only support a conjecture. $n^2 + n + 41$ is prime for $40$ values in a row and still fails at $n = 40$.

Correct: Call it a conjecture until it is proven. Say "probably true", not "true".

Mistake: Believing one counterexample is not enough to reject a conjecture

Why: A conjecture claims something about every case. If even one case fails, the claim "always" is false.

Correct: One counterexample is enough. $2$ is prime and even, so "all primes are odd" is false.

Mistake: Testing only easy, positive whole numbers

Why: Many conjectures break at $0$, at negative numbers or at fractions. "Squaring makes a number bigger" works for $2, 3, 10$ but fails for $\frac{1}{2}$: $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$.

Correct: Test special cases on purpose: $0$, $1$, negatives, fractions and large numbers.

Mistake: Confusing mathematical induction with inductive reasoning

Why: The names sound alike, but mathematical induction is a deductive proof method with a base case and an induction step.

Correct: Inductive reasoning makes a conjecture from examples. Mathematical induction proves a statement for all whole numbers.

Weather Forecasts

Forecasters compare today with many past days that looked similar. What usually happened next becomes the forecast, which is why forecasts give chances, not certainties.

If on $80$ of $100$ past days with these clouds and this wind it rained the next day, the forecast says an $80\%$ chance of rain.

Experiments in Science

Scientists repeat measurements, look for a pattern and state a rule. The rule is trusted as long as no experiment breaks it.

Hanging $1$, $2$ and $3$ equal weights on a spring stretches it $3$ cm, $6$ cm and $9$ cm. The conjecture: the stretch is $3$ cm per weight. With too many weights the spring is damaged and the rule fails.

Planning a Tournament

In a round robin tournament every team plays every other team once. Organizers can find the number of games from small cases and a conjecture.

$2$ teams play $1$ game, $3$ teams play $3$ games, $4$ teams play $6$ games. The conjecture $\frac{n(n - 1)}{2}$ fits all three cases, and here it can even be proven: each of $n$ teams plays $n - 1$ others, and every game is counted twice.

Q: If inductive reasoning can be wrong, why do we use it?

A: Because it is how we find ideas worth proving. Deductive reasoning needs a statement to start from, and that statement usually comes from noticing a pattern in examples.

Q: How many examples do I need before I can trust a conjecture?

A: No number of examples is enough for certainty. $n^2 + n + 41$ works $40$ times and then fails. More examples, and varied ones, make a conjecture more believable, but only a proof makes it certain.

Q: Is mathematical induction a kind of inductive reasoning?

A: No. Despite the name, mathematical induction is a deductive proof: you prove a base case and prove that each case implies the next, so the statement is certain for all whole numbers.

Q: Are there famous conjectures nobody has proven yet?

A: Yes. Goldbach's conjecture says every even number greater than $2$ is the sum of two primes, like $10 = 3 + 7$. Computers have checked it for every even number up to $4 \times 10^{18}$, but it is still unproven.