Fun Math Facts to Pick at Random, Each With a Way to Check It

Press Pick 5 new math facts for new facts, each with the working that shows it is true.

Fun Math Facts to Pick at Random, Each With a Way to Check It

All 104
  1. The 1,089 trick: reverse, subtract, reverse, add, and the answer is always 1,089Pick a 3-digit number whose first and last digits differ by 2 or more, such as 732. Reverse it and take the smaller away from the larger: 732 − 237 = 495. Now add the reverse of that: 495 + 594 = 1,089.
  2. The opposite faces of a die add up to 71 is opposite 6, 2 is opposite 5 and 3 is opposite 4. Dice found at Mohenjo-daro in the Indus Valley, about 4,000 years old, already followed the same rule.
  3. Every even number bigger than 2 seems to be the sum of two primes, but nobody has proved it8 = 3 + 5, 20 = 7 + 13, 100 = 47 + 53. Christian Goldbach suggested it in a letter in 1742. Computers have checked it past 4,000,000,000,000,000,000.
  4. A pizza with radius z and thickness a has a volume of pi × z × z × aIt is the ordinary formula for a cylinder: pi times the radius squared times the height.
  5. You cannot divide by zero8 ÷ 2 = 4 because 4 × 2 = 8. For 8 ÷ 0 you would need a number that gives 8 when multiplied by 0, and there is none.

Next step Practice the times tables behind the patternMake a worksheet to warm up for the trick

All 104 math facts

Number patterns 27

  • 1 is not a prime number A prime has exactly two factors, 1 and itself. The number 1 has only one factor, so it does not count.
  • 3³ + 4³ + 5³ = 6³ 27 + 64 + 125 = 216, and 6 × 6 × 6 = 216. The same three numbers give 3² + 4² = 5², because 9 + 16 = 25.
  • 12 × 12 = 144, and backwards it still works: 21 × 21 = 441 The same happens with 13 × 13 = 169 and 31 × 31 = 961.
  • 37 × 3 = 111, 37 × 6 = 222 and 37 × 9 = 333 Keep going up the 3 times table and the pattern holds all the way to 37 × 27 = 999.
  • 73 is the 21st prime number, and its mirror image 37 is the 12th There is more: 7 × 3 = 21. Count the primes from 2 to check it.
  • 2,520 is the smallest number that every number from 1 to 10 divides into exactly 2,520 ÷ 7 = 360 and 2,520 ÷ 9 = 280. No smaller number works for all ten.
  • 142,857 × 2 = 285,714, the same six digits in a new order Times 3 is 428,571, times 4 is 571,428, times 5 is 714,285 and times 6 is 857,142. Times 7 is 999,999. The digits come from 1 ÷ 7 = 0.142857142857...
  • 12,345,679 × 9 = 111,111,111 Look closely: the first number has no 8. Multiply it by 18 and you get 222,222,222.
  • 111,111,111 × 111,111,111 = 12,345,678,987,654,321 The digits climb from 1 to 9 and back down. Start small to see why: 11 × 11 = 121 and 111 × 111 = 12,321.
  • A chessboard has 204 squares on it, not 64 Count the 64 small squares, then the 49 squares that are 2 by 2, the 36 that are 3 by 3, and so on up to the 1 big board. 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204.
  • A square number never ends in 2, 3, 7 or 8 Square any whole number and the last digit is always 0, 1, 4, 5, 6 or 9. Check 1 × 1 up to 10 × 10 and you have seen every case.
  • Add up the first odd numbers and you always get a square number 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. The first 10 odd numbers add up to 10 × 10 = 100.
  • Count the spirals in a sunflower head or on a pine cone and you usually get Fibonacci numbers The seeds and scales form two sets of spirals, one turning each way. It does not happen in every plant, and sometimes the counts are multiples of Fibonacci numbers instead.
  • Divide a Fibonacci number by the one before it and you get closer and closer to 1.618 89 ÷ 55 = 1.61818 and 144 ÷ 89 = 1.61798. The number they home in on is called the golden ratio.
  • Each row of Pascal's triangle adds up to double the row above The rows 1, then 1 1, then 1 2 1, then 1 3 3 1 add up to 1, 2, 4 and 8. Each number in the triangle is the sum of the two above it.
  • Every even number bigger than 2 seems to be the sum of two primes, but nobody has proved it 8 = 3 + 5, 20 = 7 + 13, 100 = 47 + 53. Christian Goldbach suggested it in a letter in 1742. Computers have checked it past 4,000,000,000,000,000,000.
  • Every whole number can be made by adding four square numbers or fewer 31 = 25 + 4 + 1 + 1 and 23 = 9 + 9 + 4 + 1. This one is solved: Joseph-Louis Lagrange proved it in 1770, about 150 years after it was first written down.
  • In the Fibonacci sequence, every number adds the two before it: 1, 1, 2, 3, 5, 8, 13, 21 Leonardo of Pisa, known as Fibonacci, wrote about it in 1202. Indian scholars studying the rhythm of poetry had described the same numbers centuries earlier.
  • Multiply any whole number by 9 and the digits of the answer add up to a multiple of 9 Try 9 × 47 = 423. Then 4 + 2 + 3 = 9. It works for every number you pick.
  • Nobody knows whether twin primes go on forever Twin primes are primes that are 2 apart, such as 11 and 13 or 41 and 43. Mathematicians have proved there are endlessly many pairs of primes no more than 246 apart, but not yet 2.
  • The numbers from 1 to 100 add up to 5,050 Pair 1 with 100, 2 with 99, 3 with 98 and so on. That makes 50 pairs of 101. An old story, which may not be true, says Carl Friedrich Gauss spotted this as a schoolboy.
  • The only even prime number is 2 Every other even number can be divided by 2, so it has more than two factors.
  • The sieve of Eratosthenes finds prime numbers by crossing out Write the numbers 2 to 100. Circle 2 and cross out its multiples, then do the same for 3, 5 and 7. The 25 numbers left standing are all the primes under 100.
  • The smallest perfect number is 6, because 1 + 2 + 3 = 6 A perfect number equals the sum of its smaller factors. The next three are 28, 496 and 8,128. Nobody knows whether an odd perfect number exists.
  • Some cicadas spend exactly 13 or 17 years underground, and both are prime numbers One idea, still not proven, is that a prime number of years makes it harder for animals that eat them to match the cycle.
  • Two triangular numbers next to each other always add up to a square number The triangular numbers are 1, 3, 6, 10, 15, 21. Try 3 + 6 = 9, 6 + 10 = 16 and 10 + 15 = 25.
  • The year 2002 was a palindrome, and the next one is 2112 A palindrome reads the same forwards and backwards. Before 2002 the last one was 1991, only 11 years earlier.

Tricks to try 9

  • The 1,089 trick: reverse, subtract, reverse, add, and the answer is always 1,089 Pick a 3-digit number whose first and last digits differ by 2 or more, such as 732. Reverse it and take the smaller away from the larger: 732 − 237 = 495. Now add the reverse of that: 495 + 594 = 1,089.
  • A 3 by 3 magic square adds up to 15 in every row, column and diagonal Put 5 in the middle, the even numbers in the corners and the odd numbers between them. The Chinese Lo Shu square is this same pattern, and a legend says it was first seen on the shell of a turtle.
  • A Möbius strip has just one side and one edge Give a strip of paper a half twist and tape the ends. Draw a line down the middle and you return to the start having covered both faces. Two German mathematicians, Möbius and Listing, each described it in 1858.
  • A number divides by 3 if its digits add up to a multiple of 3 4,827: 4 + 8 + 2 + 7 = 21, and 21 is in the 3 times table, so 4,827 ÷ 3 = 1,609 exactly.
  • A triangle with sides of 3, 4 and 5 always has a square corner 3 × 3 + 4 × 4 = 5 × 5, which is 9 + 16 = 25. Mark a loop of string into 12 equal parts, pull it into sides of 3, 4 and 5, and test the corner against the corner of a book.
  • Cut a Möbius strip down the middle and you get one long loop, not two The new loop is twice as long and has two sides. Cut that loop down the middle again and you get two loops linked together.
  • Every 4-digit number falls into 6,174 within 7 steps Use any 4 digits that are not all the same. Arrange them largest to smallest, then smallest to largest, and subtract. Repeat. 3,524 gives 5,432 − 2,345 = 3,087, then 8,352, then 6,174. It is named after the Indian mathematician D. R. Kaprekar.
  • Halve even numbers, triple odd numbers and add 1, and you always seem to reach 1 Start at 27 and it takes 111 steps, climbing as high as 9,232 on the way. Computers have tested enormous numbers of starting values, but nobody has proved it always works. It is called the Collatz conjecture.
  • To multiply a 2-digit number by 11, add its digits and put the total in the middle 11 × 43: 4 + 3 = 7, so the answer is 473. If the total is 10 or more, carry the 1: 11 × 78 = 858.

Huge numbers 13

  • 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 seconds is exactly 6 weeks The product is 3,628,800 seconds. Divide by 60, by 60 and by 24 and you get 42 days.
  • A googol is 1 followed by 100 zeros The name was made up by Milton Sirotta, aged nine, in 1920, whose uncle was the American mathematician Edward Kasner.
  • A googolplex is 1 followed by a googol zeros, and it cannot be written out The astronomer Carl Sagan estimated that writing it in full would take more space than there is in the known universe.
  • A million seconds is about 11 and a half days, but a billion seconds is nearly 32 years 1,000,000 ÷ 86,400 seconds in a day = 11.6 days. A billion is a thousand times bigger: about 11,574 days.
  • A shuffled deck of 52 cards can be in more orders than there are seconds since the universe began The number of orders is 52 × 51 × 50 and so on down to 1. It has 68 digits and starts 80,658. A well shuffled deck has almost certainly never been in that order before.
  • Double a penny every day for 30 days and on day 30 you get $5,368,709.12 Day 1 is 1 cent, day 2 is 2 cents, day 3 is 4 cents. Day 30 is 2 multiplied by itself 29 times, which is 536,870,912 cents.
  • If you could fold a sheet of paper in half 42 times, it would be thick enough to reach the Moon Paper is about 0.1 mm thick. Doubling that 42 times gives about 440,000 km, and the Moon is about 384,000 km away. In real life the record for folding one piece of paper in half is 12 times.
  • In the United States a billion has 9 zeros, a trillion has 12 and a centillion has 303 In the older long scale, still used in many languages, a billion means a million million, which has 12 zeros.
  • Infinity is not a number, and its symbol ∞ dates from 1655 The English mathematician John Wallis introduced the sign and never explained why he chose it. The shape is called a lemniscate.
  • The largest known prime number has 41,024,320 digits It is 2 multiplied by itself 136,279,841 times, minus 1. It was found in October 2024 by a volunteer computing project and was still the record when this list was checked in September 2026.
  • One grain of wheat on the first chessboard square, doubling each square, makes 18,446,744,073,709,551,615 grains That is 2 to the power 64, minus 1. The old story of the inventor of chess asking a king for this reward shows how fast doubling grows.
  • There are 6,670,903,752,021,072,936,960 ways to fill in a sudoku grid That counts finished 9 by 9 grids that follow the rules. Treat turned, flipped and relabeled grids as the same and there are still 5,472,730,538 truly different ones.
  • There are exactly as many square numbers as there are whole numbers Pair 1 with 1, 2 with 4, 3 with 9, 4 with 16, and no number on either side is ever left out. Galileo wrote about this puzzle in 1638.

Shapes and space 15

  • A circle holds more area than any other shape with the same distance around it With 40 cm of string, a square encloses 100 square cm. A circle made from the same string encloses about 127.
  • A circle is not the only shape that rolls smoothly between two rulers A Reuleaux triangle, a triangle with bulging sides, is the same width whichever way you turn it. Some countries use seven-sided versions of this idea for coins, so that machines can measure them.
  • A pizza with radius z and thickness a has a volume of pi × z × z × a It is the ordinary formula for a cylinder: pi times the radius squared times the height.
  • A rope around the Earth needs only 6.28 extra meters to float 1 meter above the ground all the way around The extra length is 2 × pi × the extra height, whatever the size of the ball. It is the same 6.28 meters for a rope around a basketball.
  • A shape with a million sides is called a megagon Drawn as big as the Earth's equator, each side would be about 40 meters long and it would still look like a perfect circle. A chiliagon has 1,000 sides and a myriagon has 10,000.
  • About 2,200 years ago, Eratosthenes measured the Earth using shadows At midday in midsummer the Sun cast no shadow in Syene, but it made an angle of about 7 degrees in Alexandria. That is 1/50 of a circle, so the Earth had to be 50 times the distance between the cities. His answer was close to the real 40,000 km.
  • The angles inside any triangle add up to 180 degrees Tear the three corners off a paper triangle and line them up. They make a straight line. Each extra side adds another 180, so a four-sided shape has 360.
  • Bees got it right: hexagons split a flat surface into equal cells using the least wall The Roman writer Varro wondered about this in 36 BC. It was finally proved in 1999.
  • The classic soccer ball pattern is made of 12 pentagons and 20 hexagons Flattened out it is a solid called a truncated icosahedron, with 32 faces, 90 edges and 60 corners.
  • Cut a sheet of A4 paper in half and both halves are the same shape as the sheet you started with The long side is the short side times the square root of 2, about 1.414. No other ratio does this. Letter paper does not.
  • For a cube, corners minus edges plus faces equals 2, and the same is true of every solid with flat faces and no holes A cube has 8 corners, 12 edges and 6 faces: 8 − 12 + 6 = 2. A square-based pyramid gives 5 − 8 + 5 = 2. This is Euler's formula.
  • Four colors are enough to color any map so that no two neighbors match A student named Francis Guthrie noticed it in 1852. It was proved in 1976, and it was the first major theorem proved with the help of a computer.
  • Only three regular shapes can tile a flat floor on their own: triangles, squares and hexagons The corners meeting at a point must add up to exactly 360 degrees. Six triangles, four squares or three hexagons do it. Regular pentagons leave a gap.
  • The opposite faces of a die add up to 7 1 is opposite 6, 2 is opposite 5 and 3 is opposite 4. Dice found at Mohenjo-daro in the Indus Valley, about 4,000 years old, already followed the same rule.
  • There are only five Platonic solids They are solids whose faces are all the same regular shape, with 4, 6, 8, 12 and 20 faces. Dice for board games come in all five. The proof that there are no others is in Euclid's Elements.

Words and symbols 13

  • A dozen is 12, a gross is 144 and a baker's dozen is 13 A gross is a dozen dozens. Bakers in medieval England could be punished for selling short weight, so they added an extra loaf to be safe.
  • The equals sign was invented in 1557 by a Welshman, Robert Recorde He was tired of writing out is equal to. He chose two parallel lines of the same length because, he wrote, no two things can be more equal.
  • In English, forty is the only number whose letters are in alphabetical order F, o, r, t, y. It only works in English. The word one does the opposite: its letters run backwards through the alphabet.
  • In English, four is the only number that has the same number of letters as its value Count them: f, o, u, r. No other English number name manages it.
  • In English, you can count from zero to nine hundred ninety-nine without ever using the letter a The first a turns up in one thousand. That is with American number names, which leave out the word and. The letter b does not appear until one billion.
  • The largest number you can write in standard Roman numerals is 3,999 It is MMMCMXCIX. Beyond that the Romans needed extra marks, such as a bar over a letter.
  • Percent means out of every hundred It comes from the Latin per centum. So 25 percent is 25 out of every 100, which is one quarter.
  • The plus and minus signs first appeared in print in 1489 They are in a German book by Johannes Widmann, where they marked too much and too little in business sums. The division sign ÷ was first used for dividing in 1659.
  • The symbol π for pi was first used in 1706 by William Jones, a Welsh mathematician Leonhard Euler used it in his books a few decades later, and then it spread everywhere.
  • The word algebra comes from Arabic, and it once meant setting broken bones Al-jabr is in the title of a book written around 825 by the Persian mathematician al-Khwarizmi. The word algorithm comes from his name.
  • The word calculate comes from the Latin for pebble Romans did sums by moving small stones, called calculi, on a counting board.
  • The word zero comes from the Arabic sifr, which meant empty It traveled through Italian as zefiro and then zero. The word cipher comes from the same root.
  • Zero cannot be written in Roman numerals The Romans had no symbol for it. Medieval scholars wrote the Latin word nulla, meaning none, or just the letter N.

Math history 7

  • A year is a leap year if it divides by 4, except century years, which must divide by 400 So 2000 was a leap year, but 1900 was not and 2100 will not be. That makes 97 leap years in every 400 years.
  • The digits 0 to 9 that we use every day were invented in India Indian mathematicians developed the system from the 1st to the 4th century. Arabic scholars adopted it by the 9th century, and it reached Europe through them.
  • Graph theory began in 1736 with a puzzle about seven bridges People in Königsberg wondered whether a walk could cross each of the city's seven bridges exactly once. Leonhard Euler proved that it could not.
  • The Maya of Central America used zero too, and worked it out on their own They needed it for their calendar, which counted in twenties.
  • Pi Day is March 14, because the date is written 3/14 It began in 1988 at a science museum in San Francisco, where staff marched in a circle and then ate fruit pies. UNESCO has since made it the International Day of Mathematics.
  • There is no Nobel Prize for mathematics The best known prize is the Fields Medal. It is given every four years to two, three or four mathematicians who are under 40.
  • We have 60 minutes in an hour because the Sumerians and Babylonians counted in sixties 60 is handy because it divides exactly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. The 360 degrees in a circle come from the same system.

Pi and special numbers 11

  • 0.999 repeating forever is exactly equal to 1 One third is 0.333 repeating. Multiply both by 3 and you get 1 = 0.999 repeating.
  • Any number except zero raised to the power 0 equals 1 Go down the powers of 2: 8, 4, 2. Each step divides by 2, so the next one, 2 to the power 0, is 1.
  • The digits of pi never end and never settle into a repeating pattern It starts 3.14159. Johann Lambert proved in the 1760s that pi cannot be written as a fraction.
  • The fraction 22/7 is close to pi, but 355/113 is far closer 22 ÷ 7 = 3.142857, right to 2 decimal places. 355 ÷ 113 = 3.1415929, right to 6 decimal places.
  • In a one-lap race, each lane starts ahead of the lane inside it by the lane width times 2 × pi It is the rope-around-the-Earth fact at work: one lane further out adds the same extra distance around the bends whatever the size of the track.
  • In binary, the way computers count, the number 2 is written 10 Binary has only the digits 0 and 1. Counting goes 1, 10, 11, 100, 101, and 100 in our numbers is 1100100.
  • Minus 40 degrees is the same temperature in Celsius and Fahrenheit Multiply −40 by 9, divide by 5 and add 32, and you land back on −40. It is the only place the two scales meet.
  • NASA steers spacecraft between planets using only 15 decimal places of pi The value is 3.141592653589793. For a circle 40 billion miles across, that is accurate to about half an inch.
  • The square root of 2 cannot be written as a fraction It starts 1.41421356 and, like pi, never repeats. It is the length of the diagonal of a square with sides of 1.
  • You cannot divide by zero 8 ÷ 2 = 4 because 4 × 2 = 8. For 8 ÷ 0 you would need a number that gives 8 when multiplied by 0, and there is none.
  • Zero is an even number An even number is one you can split into two equal whole parts, and 0 = 0 + 0. It also sits between two odd numbers, −1 and 1.

Chance and puzzles 9

  • 7 is the most likely total when you roll two dice There are 36 ways two dice can land, and 6 of them make 7. Only 1 way makes 2, and only 1 makes 12.
  • Every month beginning on a Sunday has a Friday the 13th If the 1st is a Sunday, the 8th is a Sunday too, so the 13th is five days later: a Friday.
  • Flip a coin 10 times and there are 1,024 different ways it can go That is 2 multiplied by itself 10 times. Ten heads in a row is just 1 of those 1,024.
  • The hands of a clock overlap only 22 times a day, not 24 They meet 11 times in every 12 hours. After 12:00 the next meeting is at about 1:05, then 2:11, and the one you would expect near 11:55 does not happen until 12:00.
  • If 30 people all shake hands with each other, that is 435 handshakes Each of the 30 shakes 29 hands, which is 870, but that counts every handshake twice. 870 ÷ 2 = 435.
  • In a group of just 23 people, two of them more likely than not share a birthday The chance is about 50.7 percent. In a class of 30 it rises to about 70.6 percent. Try it with your class.
  • In many real-life lists of numbers, about 30 percent of them start with the digit 1 Only about 5 percent start with 9. It works for river lengths, populations and house prices, and it is called Benford's law. An astronomer first noticed it in 1881 because the early pages of his book of number tables were the most worn.
  • In the three doors puzzle, switching doors doubles your chance of winning You pick 1 door of 3. The host, who knows where the prize is, opens an empty door and offers a swap. Sticking wins 1 time in 3 and swapping wins 2 times in 3. It is known as the Monty Hall problem.
  • It takes about 7 riffle shuffles to mix a deck of cards properly Two mathematicians worked this out in a paper published in 1992. After 5 shuffles a deck is only starting to be random.

How it works

These fun math facts come five at a time, and each has a line underneath that shows how to check it. Press Pick 5 new math facts, or choose a group such as Huge numbers from the menu that reads All math facts. All 104 were checked by calculation or against a named source, so you can put one on the board without wondering whether it is a myth.

Read the full guide

A fact, then the working

The page opens on one fact. Under it is the part most lists leave out: the sum, the example or the date that lets you test the claim. Copy takes one fact with its working. The menu that reads All math facts narrows the pick to one of 10 groups. Nothing repeats until you have seen all 104.

How each fact was checked

Anything that is arithmetic was worked out twice: the digit patterns, the 204 squares on a chessboard, the 68 digits in the number of ways to order a deck of cards. History and records were checked against Wikipedia and NASA in September 2026, and the largest known prime is dated because it will be beaten. Several weird math facts that circulate online did not survive. Paper can be folded in half more than 8 times, for instance: the record is 12. Where a story is only a story, such as young Gauss adding 1 to 100, the check line says so.

Turning a fact into five minutes of math

A fact earns its place in a lesson when pupils can test it. Read the fact out before you show the board, and ask whether it sounds true. The 1,089 trick and the 6,174 routine need only subtraction, and every pupil starts from a different number and lands on the same answer. The birthday fact can be tested on your own class list. For interesting math facts with no sum to do, such as who invented the equals sign, ask pupils to guess the century first.

  • Choose Tricks to try from the All math facts menu when you have a spare five minutes. Each one needs only pencil and paper.
  • After a times table pattern such as 37 × 3 = 111, send pupils to the multiplication flash cards to practice that table.
  • With kindergarten to second grade, stay with small numbers: the dice faces that add up to 7, the odd numbers that build squares, the 4 letters in four.
  • After one example works, ask whether it always works. Goldbach's idea has been tested past 4 quintillion and is still not proved.

Questions and answers

Are these math facts actually true?

Yes, as far as careful checking can make them. Every calculation was done twice, and every date, name and record was read in a source on September 18, 2026. Claims that could not be confirmed were left out, which is why some amazing math facts you may have seen elsewhere are missing. If you find a mistake, please report it through the contact page.

Which are good fun math facts for kids in elementary school?

Start with facts a child can test alone. In English, four is the only number with as many letters as its value. Opposite faces of a die add up to 7. A chessboard holds 204 squares once you count the big ones. Older pupils enjoy the doubling facts, such as the penny that passes 5 million dollars on day 30.

Which of these are mind blowing math facts?

It depends on the age of the audience. For most classes it is the deck of cards: a shuffled deck can be in more orders than there have been seconds since the universe began. For older students, try 0.999 repeating being exactly 1, or the pairing that shows there are as many square numbers as whole numbers.

How can I use random math facts as a lesson starter?

Put a math fact on the board as pupils come in, with one question: how could we test this? Give them three minutes. Favor the facts with a sum in the check line over the ones that are only a date.

Why do some facts say that nobody knows?

Because in mathematics a pattern is not settled until someone proves it. Twin primes, Goldbach's sums and the halve-or-triple game have held for every number ever tried, and still no proof exists. The group called Proved and unproved mixes those with two that were settled: the four color map problem in 1976 and the four squares rule in 1770.

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