50 maths myths & paradoxes

9️⃣0.999… is almost 1, but not quite 1

Nine, nine, nine, nine… still not 1? Bring a ruler. You will run out of paper first.

The maths: Let x = 0.999… Then 10x = 9.999…, so 10x − x = 9, which gives 9x = 9 and x = 1. Or: 1/3 = 0.333…, and three thirds make 1 = 0.999…. There is no number between 0.999… and 1, so they cannot be different.

Try it yourself: Ask a friend to name a number that lies between 0.999… and 1. They can't: the gap is smaller than any gap you can write down.

Test it: 0.999… → fraction in the Maths Explainers →

🚫Divide by zero and you get infinity

Your calculator says 'Error'. It is not being lazy. It is being honest.

The maths: a ÷ b = c means c × b = a. For 5 ÷ 0 we would need a number c with c × 0 = 5, but anything times 0 is 0. And the values of 1/x as x → 0 head to +∞ from the right and −∞ from the left, so there is no single answer even as a limit.

Try it yourself: Type 1 ÷ 0.1, 1 ÷ 0.01, 1 ÷ 0.001 … then 1 ÷ −0.1, 1 ÷ −0.01 … and watch the answers run off in opposite directions.

Test it: the limit of 1 ÷ x at 0 in the Calculus lab →

0️⃣Zero is neither even nor odd, it's nothing

Zero is the most even number there is. It just never gets invited to parties.

The maths: An even number is a whole number divisible by 2, and 0 ÷ 2 = 0 exactly. It is 2 × 0. Even and odd numbers alternate on the number line (… −1, 0, 1, 2 …), and the neighbours of 0 are odd, so 0 must be even.

Try it yourself: Check the rule 'even + even = even' with 0: 4 + 0 = 4 still works. Try the same test if you call zero odd: odd + odd should be even, but 1 + 0 = 1 would break it.

✖️Multiplying always makes a number bigger

Half of a half is a quarter. That is multiplication making things smaller, and pizza knows it.

The maths: Multiplying by 0.5 means 'take half', so 8 × 0.5 = 4. Multiplying by −2 doubles the size and reverses the direction: 8 × (−2) = −16. Multiplication is scaling, and scaling can shrink.

Try it yourself: Ask for 0.25 of a 200 rupee bill: 0.25 × 200 = 50. The answer is smaller than 200 even though you multiplied.

➖'Minus times minus is plus' is just a rule someone made up

It is not a rule. It is a consequence. Nobody gets to vote on it.

The maths: Start from (−3) + 3 = 0. Multiply by −2: (−2)(−3) + (−2)(3) = 0. Since (−2)(3) = −6, the number (−2)(−3) must be +6 for the sum to be 0. The distributive law needs it. On the number line, multiplying by −1 is a half-turn, and two half-turns make a full turn back to the start.

Try it yourself: Play 'the enemy of my enemy': two flips of direction bring you back facing the way you started.

🌡️−5 is bigger than −2 because 5 is bigger than 2

Anyone who has owed money knows that owing 5 is worse than owing 2.

The maths: −5 lies to the left of −2, so −5 < −2. A temperature of −5 °C is colder than −2 °C, and a debt of 5 rupees is worse than a debt of 2. Adding a positive number moves you right, so −5 + 3 = −2 shows −2 is bigger.

Try it yourself: Draw a number line from −6 to 6 and mark −5 and −2. The one on the right is the greater number.

1️⃣1 is a prime number

1 has only itself as a factor. That sounds like a prime, but it is a unit, and units are a different club.

The maths: If 1 were prime, prime factorisation would no longer be unique: 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3 … The fundamental theorem of arithmetic says every whole number greater than 1 factors into primes in exactly one way, so 1 is excluded.

Try it yourself: List the factors of 7 (1 and 7) and of 1 (just 1). A prime has exactly two.

2️⃣All prime numbers are odd

2 is the only even prime, which makes it the oddest prime of all.

The maths: 2 has exactly two factors, 1 and 2, so it is prime. Every other even number is divisible by 2 and also greater than 2, so it has a factor other than 1 and itself, which means it is not prime.

Try it yourself: Try to find another even prime. Any even number above 2 has 2 as a factor and is itself bigger than 2, so it fails.

🥧π is exactly 22 ÷ 7

22/7 is π's stunt double. Close enough for the camera, not for the crash scene.

The maths: 22/7 is a fraction, so its decimal repeats (142857 forever). π is not a fraction of two whole numbers, so its decimal never repeats. The gap is small but real: 22/7 − π ≈ 0.00126. A better fraction is 355/113 ≈ 3.14159292.

Try it yourself: Divide 22 by 7 and 355 by 113 on a calculator, then compare with the π button.

Test it: 22 ÷ 7 in the Maths Explainers →

🍕To add fractions, add the tops and add the bottoms

Half a pizza plus a third of a pizza is not two-fifths of a pizza. It is five-sixths, and it's way tastier.

The maths: Halves and thirds are pieces of different size, so you cannot add them until you cut both into sixths: 3/6 + 2/6 = 5/6. The 'add tops and bottoms' answer 2/5 is smaller than 1/2, which is impossible since you added something positive to 1/2.

Try it yourself: A quick sanity check: the sum must be bigger than each part. 2/5 = 0.4 is smaller than 1/2, so the shortcut fails.

📉Up 50% then down 50% puts you back where you started

Your share price goes up half, then down half. Your wallet goes down a quarter. Nobody warned you.

The maths: Percentages are always of the current amount. 50% of 100 is 50, giving 150, but 50% of 150 is 75, giving 75. In multiplication terms, 1.5 × 0.5 = 0.75, not 1. To undo a 50% rise you need a fall of one third.

Try it yourself: Try it with ₹200: add 50% (₹300), then take off 50% (₹150). Compare with the ₹200 you began with.

🧮(a + b)² = a² + b²

The middle term 2ab is the bit everyone forgets, and the bit the exam always tests.

The maths: (a + b)² means (a + b)(a + b). Multiply out: a·a + a·b + b·a + b·b = a² + 2ab + b². As a picture, a square of side a + b is made of an a-square, a b-square and two a-by-b rectangles. Check with a = 3, b = 4: (3 + 4)² = 49, but 9 + 16 = 25.

Try it yourself: Cut a square of side 7 into a 3 × 3 square, a 4 × 4 square and two 3 × 4 rectangles. The four pieces fill it exactly.

Test it: draw y = (x + 3)² and y = x² + 9 in the Function Explorer →

📐√(a² + b²) = a + b

If that were true, the shortcut across a field would be no shorter than walking round the edge.

The maths: Square roots do not split over addition. That is exactly why the diagonal of a right triangle (Pythagoras) is shorter than the two sides added together: it is the straight line, and the triangle inequality says a straight line is the shortest route.

Try it yourself: Walk along two sides of a rectangular room, then along its diagonal, and count your steps.

Test it: Pythagoras in the Geometry lab →

±x² = 4 means x = 2

Half the answer is still a wrong answer. Ask a quadratic how many roots it has.

The maths: Both 2² and (−2)² equal 4. Solving x² − 4 = 0 gives (x − 2)(x + 2) = 0, so x = 2 or x = −2. The symbol √4 means only the positive value 2, but the equation x² = 4 asks for every x that works.

Try it yourself: Sketch y = x² and the line y = 4: the line cuts the curve at two points, one on each side.

Test it: two zeros of a quadratic in the Function Explorer →

🧩(x − 1)(x − 2) = 6 means x − 1 = 6 or x − 2 = 6

The zero-product rule only works with 0. Six is not a zero, however hard you stare at it.

The maths: If ab = 0 then a = 0 or b = 0, because you cannot make 0 by multiplying two non-zero numbers. But many pairs multiply to 6: 2 × 3, 1 × 6, (−2) × (−3) … so nothing forces a factor to be 6. Solve x² − 3x − 4 = (x − 4)(x + 1) = 0 to get x = 4 or x = −1.

Try it yourself: Test the 'wrong' answer x = 7: (7 − 1)(7 − 2) = 30, not 6.

🪞Squaring both sides of an equation is always safe

Squaring is a one-way street: it lets in unwelcome guests who pretend to be answers.

The maths: x = −2 is not a solution of x = 2, but squaring gives x² = 4, which has −2 as a solution. Squaring merges a and −a. Example: √(x + 2) = x. Squaring gives x + 2 = x², so x = 2 or x = −1, but x = −1 gives √1 = 1 ≠ −1, so only x = 2 works.

Try it yourself: After you solve any equation that involved a square or a square root, plug the answers back in.

↔️Multiply both sides of an inequality by anything and the sign stays

Multiply by −1 and the whole world turns round. So does the inequality sign.

The maths: On the number line, multiplying by a negative number reflects it through 0, which swaps left and right. Since −4 lies to the right of −6, −4 > −6. Multiplying by a positive number does not reflect anything, so the sign stays.

Try it yourself: Start with 1 < 2. Multiply by −1: you get −1 and −2. Which is bigger now?

🎩The proof that 1 = 2 (so maths is broken)

It is a magic trick, and the disappearing act is a division by zero.

The maths: Every step until the cancellation is correct, but (a + b)(a − b) = b(a − b) is really 0 = 0, which is true whatever the numbers are. Cancelling (a − b) divides by 0, and that step is not allowed. Bogus proofs almost always hide a division by zero or a wrongly used square root.

Try it yourself: Substitute a = b = 1 into each line and see where 'true' quietly becomes 'both sides are 0'.

🟦A square is not a rectangle

Every square is a rectangle. Not every rectangle is a square, just like not every Tamil speaker is from Chennai.

The maths: A rectangle is a quadrilateral with four right angles. A square has four right angles too, so it qualifies. In the same way a square is also a rhombus (four equal sides) and a parallelogram. The shapes are nested, like boxes inside boxes.

Try it yourself: Draw a rectangle and slowly make its two different sides equal. At the moment they match, it has become a square, without ever leaving the family.

Test it: bend quadrilaterals in the Geometry lab →

🔺The angles of a triangle always add up to 180°

True on your notebook page. On a globe, the triangle has other ideas.

The maths: Take the North Pole and two points on the equator 90° of longitude apart. The triangle has three right angles, so its angles add to 270°. The 180° rule follows from Euclid's parallel postulate, which holds on flat surfaces only.

Try it yourself: Draw a triangle on an orange with a marker: two lines from the 'pole' down to the 'equator', then along the equator. Measure the corners.

Test it: the angle sum of a triangle in the Geometry lab →

⭕Bigger circles have a bigger π

π is the same for a coin and for the equator. That is what makes it a constant.

The maths: All circles are scaled copies of each other (similar figures). If you double the diameter, the circumference also doubles, so their ratio never changes. That single fixed ratio is what we call π.

Try it yourself: Wrap a string around a bangle, a plate and a dustbin lid. Measure each circumference and diameter, and divide. You will get about 3.14 each time.

Test it: circles in the Circles & Conics lab →

📏Double the sides and the area doubles

Order a 14 inch pizza instead of 7 inch and you get four times the pizza, not twice. The price will not be four times, so it's a bargain.

The maths: Area is length × length, so the scale factor is squared: 2² = 4. Volume is length × length × length, so it is cubed: 2³ = 8. This is also why big animals need thick legs: weight goes up with the cube, but the bone's cross-section only with the square.

Try it yourself: Build a 2 × 2 × 2 block from 1 × 1 × 1 cubes: it takes 8 cubes and shows 4 times the surface on each face.

Test it: resize solids in the Mensuration lab →

✈️On a flat map, a straight line is the shortest route

That is why long flights, such as Delhi to San Francisco, swing far north near the Arctic. The pilot is not lost.

The maths: A flat map stretches the curved surface of the Earth, so straight lines on a map are not shortest routes on the globe. A great circle is what you get by cutting the sphere through its centre, and the shortest path between two points lies along one.

Try it yourself: Stretch a string between two cities on a globe. The taut string is the shortest path, and it bends when you copy it onto a flat map.

🛤️Parallel lines never meet

On a flat page, never. On a globe, all the lines of longitude meet at the poles.

The maths: Two lines of longitude both cross the equator at a right angle, so they start out parallel, yet they meet at the North Pole. Euclid's parallel postulate, which says there is exactly one parallel through a point, holds on a plane but not on a sphere.

Try it yourself: On a globe, put two fingers on the equator and follow two meridians north.

🍦A cone holds half as much as a cylinder of the same size

The ice-cream cone is a bargain hunter's nightmare. It holds exactly one third of the cup next to it.

The maths: The cone's cross-sections shrink from the full base size to a point, and adding them all up gives one third of the cylinder's volume. You can check it by pouring: three cones of water fill the cylinder exactly.

Try it yourself: Make a cone and a cylinder of the same base and height from card, fill the cone with rice and pour it into the cylinder three times.

Test it: pour cones into a cylinder in the Mensuration lab →

🪵Any three lengths can make a triangle

Try building a triangle from sticks of 1, 2 and 3. You will get a very sad straight line.

The maths: The shortest way between two corners is a straight line, so one side can never be longer than the other two put together. If 1 + 2 = 3, the two short sticks lie flat along the long one. With 2, 3 and 4 they can swing apart to make a real triangle.

Try it yourself: Cut straws to 2, 3 and 6 cm and try to join them. Then try 2, 3 and 4.

Test it: drag a triangle corner onto a line in the Mensuration lab and watch the area reach 0 →

🔲Shapes with the same perimeter have the same area

A farmer with 20 metres of fence wants the biggest field. Square wins, a long thin rectangle loses.

The maths: For a fixed perimeter, the more 'round' the shape, the more area it holds. Among rectangles the square is best, and among all shapes the circle is best. That is why bubbles are round.

Try it yourself: Make a loop of 20 cm of string. Shape it into a thin rectangle, a square and a circle, then compare how much space each one covers.

Test it: area and perimeter in the Mensuration lab →

🔻Pythagoras' theorem works for every triangle

a² + b² = c² is a private club for right triangles.

The maths: When the angle C is 90°, cos C = 0 and the extra term disappears, leaving Pythagoras. For an acute angle the term is positive so c² < a² + b², and for an obtuse angle c² > a² + b². The check a² + b² versus c² tells you the kind of triangle.

Try it yourself: Test a triangle with sides 4, 5 and 6: 16 + 25 = 41, which is bigger than 36, so it has no right angle (and it is acute).

Test it: right triangles in the Trigonometry lab →

📈Every graph is the graph of a function

A circle is a perfectly good shape and a terrible function.

The maths: The circle x² + y² = 25 has, for x = 3, two points: y = 4 and y = −4. One input, two outputs, so it is not a function. Split it into two halves, y = √(25 − x²) and y = −√(25 − x²), and each half is a function.

Try it yourself: Hold a pencil vertically and sweep it across a graph. If it ever touches the graph twice at once, it is not a function.

Test it: plot functions in the Function Explorer →

🎯sin(A + B) = sin A + sin B

sin 30° + sin 60° = 1.37, but sin 90° = 1. The sine has no interest in distributing.

The maths: A function does not usually 'split' over addition. Test it: sin 30° = 0.5 and sin 60° ≈ 0.866, whose sum is 1.366, but sin 90° = 1. The true formula has cross terms, just as (a + b)² does. It can be shown with a unit-circle picture or by rotating a vector.

Try it yourself: Use a calculator with A = 30° and B = 60°, and compute both sides.

Test it: the unit circle in the Trigonometry lab →

✳️The derivative of a product is the product of the derivatives

It would make calculus much easier. Which is exactly why it isn't true.

The maths: Take f = g = x. Then f g = x², whose derivative is 2x. The myth would give f′ × g′ = 1 × 1 = 1. The product rule gives 1·x + x·1 = 2x, which is correct. Think of a rectangle with sides f and g: as both sides change, the area gains a strip along each side.

Try it yourself: Differentiate x·x by the myth and by the product rule, then compare with the known derivative of x².

Test it: derivatives in the Calculus lab →

🚧A limit is a value the function never reaches

The limit is where the function is heading, not always where it isn't.

The maths: For f(x) = 5 (a constant), every value equals its limit 5. For f(x) = sin(x) ÷ x, the value at x = 0 is missing, but the limit is 1. For f(x) = (sin x) ÷ x again, as x → ∞ the values cross the limit 0 infinitely often. So 'never reaches' is not part of the definition.

Try it yourself: Draw the graph of y = sin(x) ÷ x for large x and see it wiggle across y = 0 while heading for it.

Test it: limits in the Calculus lab →

⛰️If f′(x) = 0 then f has a maximum or a minimum there

The curve stops climbing for a second, then carries on climbing. Not every pause is a summit.

The maths: For f(x) = x³, f′(x) = 3x² is 0 at x = 0, yet the curve keeps rising on both sides. To decide, check whether f′ changes sign (from + to − is a maximum, from − to + is a minimum) or use the second derivative test.

Try it yourself: Plot y = x³ and y = x². Both have zero slope at 0. Only one has a valley there.

Test it: maxima and minima in the Calculus lab →

🔔A continuous function must be smooth

You can draw |x| without lifting your pencil, but you will get a sharp corner in your hand.

The maths: At x = 0 the slope approaching from the left is −1 and from the right is +1. They disagree, so there is no single tangent. Differentiable implies continuous, but not the other way round.

Try it yourself: Try to draw one tangent line at the point of V on the graph of y = |x|. Which one would you choose?

Test it: corners and continuity in the Calculus lab →

🌊An integral is always an area, and areas are positive

The integral of sin x from 0 to 2π is 0. The area there is certainly not zero.

The maths: From 0 to π, sin x is above the axis with area 2; from π to 2π it is below with area −2. They cancel to give 0. If you want the total (unsigned) area, split the interval where the curve crosses the axis and add the absolute values: 2 + 2 = 4.

Try it yourself: Cut out the two humps of one sine wave from paper. They are the same size, so they weigh the same.

Test it: signed and total area in the Calculus lab →

🚀x¹⁰⁰ eventually grows faster than 2ˣ

The polynomial gets a head start, and the exponential lets it, and then it overtakes it forever.

The maths: For a while x¹⁰⁰ is far bigger than 2ˣ. But 2ˣ doubles every step, while x¹⁰⁰ only grows by the factor (1 + 1/x)¹⁰⁰, which tends to 1. Beyond a certain x (roughly 1000), 2ˣ wins for good. That is the difference between polynomial and exponential growth.

Try it yourself: Compare x² and 2ˣ: at x = 3 you get 9 and 8, at x = 4 you get 16 and 16, at x = 5 you get 25 and 32.

Test it: compare growth in the Function Explorer →

🪙After five heads in a row, tails is due

The coin has no memory. It has never heard of you.

The maths: The gambler's fallacy mistakes the law of large numbers for a force that corrects the past. Over thousands of flips the proportion of heads gets close to 1/2, but not because the coin compensates: the early streak just gets diluted by all the later flips.

Try it yourself: Flip a coin 20 times and write down the longest run. Runs of 4 or 5 are normal.

Test it: flip coins by the thousand in the Probability lab →

🎲Flip a fair coin 10 times and you'll get exactly 5 heads

Exactly 5 heads happens about a quarter of the time. The coin is not being unfair, just being random.

The maths: There are 2¹⁰ = 1024 equally likely outcomes, and C(10, 5) = 252 of them have exactly 5 heads, so P = 252 ÷ 1024 ≈ 0.246. Results like 4 or 6 heads are just as ordinary. The proportion gets closer to one half as you flip more, but the exact count does not.

Try it yourself: Flip a coin 10 times, ten times over. Count how many of the ten rounds gave exactly 5 heads.

Test it: the binomial curve in the Probability lab →

💰The average tells you what is typical

When a billionaire walks into a bar, the average person in the bar becomes a millionaire.

The maths: Five salaries: ₹20k, ₹22k, ₹25k, ₹28k and ₹500k. The mean is ₹119k, which nobody earns. The median is ₹25k, which describes most of the group. Whenever the data has outliers or is skewed (incomes, house prices), quote the median.

Try it yourself: Ask the class for their pocket money, then add the richest person's amount as 100 times bigger. See what happens to the mean and to the median.

Test it: drag data points in the Probability lab →

🎂You need 183 people for a shared birthday to be likely

In a class of 23 there is a better than even chance that two people share a birthday. The birthday cake budget is at risk.

The maths: Count pairs, not people: 23 people make 253 different pairs, and each pair has a 1/365 chance of matching. Precisely, P(no match) = (365/365)(364/365)…(343/365) ≈ 0.493, so P(match) ≈ 0.507. With 70 people it is over 99.9%.

Try it yourself: Check the birthdays of everyone in your class, or of the players in two football teams (22 players).

👶A family has two children and at least one is a boy: the other is a boy with probability 1/2

It feels like a coin flip, but the sample space says otherwise.

The maths: The equally likely families are BB, BG, GB and GG. 'At least one boy' rules out GG and leaves BB, BG, GB, three cases, only one of which is BB. If you instead know 'the older child is a boy', the cases are BB and BG, so the chance is 1/2. The wording matters.

Try it yourself: Write the four possible two-child families and cross out the ones the clue forbids.

🚪Monty Hall: with two doors left it's 50–50, so switching is pointless

Even famous mathematicians got this one wrong in public. You are in good company.

The maths: Your first pick is right 1/3 of the time and wrong 2/3 of the time. If it was wrong (2/3), the host is forced to open the only other losing door, so the remaining door has the prize. Switching therefore wins whenever your first pick was wrong: 2/3.

Try it yourself: Play it with three cups and a coin under one of them. Play 30 rounds of 'always stay', then 30 rounds of 'always switch'.

Test it: simulate draws in the Probability lab →

🍦If two things go up together, one causes the other

Ice cream sales and drowning both rise in summer. Ban ice cream? No. It's hot weather.

The maths: Ice-cream sales and drownings both rise in summer because of hot weather, which drives both. Correlation only measures how well two sets of data move together on a line. To show causation you need a controlled experiment or a good causal argument.

Try it yourself: Find your own funny example: number of umbrellas sold and number of colds could both rise in the rainy season.

🎟️1-2-3-4-5-6 is less likely to win the lottery than a 'random' set

Every combination is equally unlikely, including the one you would never pick.

The maths: In a 6-out-of-49 lottery there are C(49, 6) = 13,983,816 equally likely sets, and 1-2-3-4-5-6 is one of them, just like your birthday numbers. The 'random-looking' sets are more numerous as a class, but each single set is no likelier. The tip: if you win, so will everyone who picked the popular pattern, so pick unusual numbers to avoid sharing the prize.

Try it yourself: Use dice or a random-number app to generate a set, then write the pattern next to it. Both are one line in the 13,983,816.

🩺A 99% accurate test says positive: you're 99% likely to have the disease

The test is right 99% of the time, and yet most positive results are false alarms. Welcome to base rates.

The maths: Take 100,000 people. 100 are ill and 99 of them test positive. Of the 99,900 healthy people, 1% (999) also test positive. So of 1098 positive tests, only 99 are real: 99 ÷ 1098 ≈ 9%. This is Bayes' theorem: rare conditions produce mostly false positives.

Try it yourself: Draw the 100,000 people as a tree with four branches (ill/healthy, positive/negative) and count.

♾️Infinity is the biggest number

Infinity plus one is still infinity. Also infinity times infinity. This is the fun part.

The maths: If ∞ were a number, ∞ + 1 = ∞ would give 1 = 0 after subtracting ∞. Expressions like ∞ − ∞ or ∞ ÷ ∞ have no fixed value. In calculus, ∞ shows up as 'grows without bound', in the language of limits.

Try it yourself: Ask: what is the biggest whole number? Whatever you say, add 1. There is no biggest, which is the idea of infinity.

🧮There are fewer even numbers than whole numbers

Hilbert's hotel is full, but a new guest can always be given a room. This is not a hotel review.

The maths: Pair every whole number n with the even number 2n: 1 ↔ 2, 2 ↔ 4, 3 ↔ 6 … Every whole number gets exactly one partner, and every even number is used exactly once. Two collections with a perfect pairing have the same size. Galileo noticed the same thing with squares.

Try it yourself: Write the whole numbers in a row and their doubles in a row beneath them, and try to find one that is left over.

🧱All infinities are the same size

Some infinities are bigger than others. Cantor proved it, and then got into trouble for it.

The maths: Cantor's diagonal argument: suppose you could list all decimals between 0 and 1. Build a new decimal whose first digit differs from the first decimal in your list, whose second digit differs from the second, and so on. It is not the same as any number on the list, so the list was incomplete. The whole numbers, fractions and even-numbers can be listed, but the real numbers cannot.

Try it yourself: Write a list of any five decimals and use the diagonal trick to build a sixth that isn't in the list.

🐢Achilles can never overtake the tortoise

Zeno said motion is impossible. He never had to catch a bus.

The maths: Let Achilles run at 10 m/s and the tortoise at 1 m/s, with a 90 m head start. The gaps are 90 m, 9 m, 0.9 m … and the times are 9 s, 0.9 s, 0.09 s … These form a geometric series with sum 9 ÷ (1 − 0.1) = 10 seconds. Adding infinitely many shrinking pieces can give a finite total.

Try it yourself: Add 1/2 + 1/4 + 1/8 + … on a calculator. It creeps towards 1 and never goes past it.

Test it: a geometric series settling on its sum in the Maths Explainers →

🔢If a pattern works for the first 40 cases, it's proven

n² + n + 41 makes primes for 40 numbers in a row. Then it makes 1681 = 41 × 41. Examples are not proofs.

The maths: n² + n + 41 is prime for every n from 0 to 39. At n = 40 it equals 40² + 40 + 41 = 1681 = 41 × 41, which is not prime. A proof works for all n at once, using reasoning (like induction) instead of testing cases. A single counterexample, on the other hand, is enough to disprove a claim.

Try it yourself: Test n² + n + 41 for n = 0, 1, 2, 3 … until you reach 40. Then look at the result.