∞ Grade 6
For curious 11 and 12 year olds. Math that is just around the corner from what you do at school. One sheet a night, twenty minutes, and a few things that will surprise you.
Sharpen school math, one notch harder
Everything on the grade 6 curriculum, with the "why" that school sometimes skips and problems a bit harder than the ones you'll be set.
Fractions that fight back
Why flip-and-multiply works, and how the Egyptians wrote fractions.
Ratios, rates and recipes
Scaling, speed, and spotting the better deal.
Percentages in your head
8% of 25 is the same as 25% of 8. Use it.
Below zero
Why a negative times a negative is positive, properly.
Multiply like a magician
23 × 17 in two seconds using a square.
Gears, lockers and common factors
GCF, LCM, Euclid's algorithm and the 1000 lockers.
Why elephants have thick legs
Area, volume, and what happens when you double everything.
Lying with averages
Mean, median, and how a billionaire wrecks both.
The balance
Equations, inequalities, and think-of-a-number tricks.
Big and tiny numbers
Exponents, scientific notation, and the distance to the stars.
Why 1/7 goes round in circles
Repeating decimals, and why every fraction has to repeat.
Area by counting dots
Polygons on a grid, and a formula that feels like cheating.
Three ways to multiply
Lattice, Russian peasant, and the 90s trick. All secretly the same.
How many piano tuners are in Chicago?
Estimating anything from almost nothing.
How to lie with a graph
Chopped axes, swollen pictures, and how to spot them.
Why PEMDAS
And the internet's favourite argument, settled.
The 125-million-dollar unit mistake
Converting units like algebra, and the spacecraft that didn't.
Around the corner new ideas
Things you will meet in a few years, or never at school at all. None of them need anything beyond what you already know. The cards call these "Corner" for short.
Primes and the Sieve
The atoms of arithmetic, and the fast test for whether a number is prime.
Counting to 31 on one hand
Binary, and a mind-reading trick built from it.
Clock arithmetic
Remainders, the last digit of 3100, and casting out nines.
Pascal's triangle
Paths, choices, powers of two, and a hidden fractal.
Gauss and the lazy sum
1 + 2 + … + 100 in your head, and why odd numbers make squares.
Fibonacci and the golden ratio
Stairs, rabbits, sunflowers, and 1.618.
Pythagoras and square roots
A proof you can see, and a number that isn't a fraction.
Imaginary numbers are real
The square root of minus one is a quarter turn.
Matrices: grids that do things
Four numbers that rotate, stretch and flip the plane.
The pigeonhole principle
Ten pigeons, nine holes, and why two people in London have the same number of hairs.
Odd, even and things that never change
The chessboard puzzle, and a coin trick done blindfolded.
Hilbert's hotel
Full, and still room for more. The sheet the club is named after.
Seven bridges of Königsberg
The puzzle that invented graph theory.
Dice don't lie
Why 7 wins, and the bet that quietly lost a gambler money.
Nim: the game you cannot lose
A winning strategy hidden in binary.
Secret codes and how to break them
Caesar shifts, clock arithmetic, and counting letters.
The detective's method for sequences
Find the formula from the differences. And one pattern that lies.
How many ways
Factorials, a shuffle nobody has ever seen, and the birthday problem.
Measuring the Earth with a stick
How Eratosthenes did it in 240 BC, and how you can measure a tree.
The Möbius strip
Paper, tape, scissors, and a shape with one side.
Counting digits
Logarithms before anyone tells you about them. Decibels, earthquakes, stars.
Why a pizza is πr²
Slice it, rearrange it, and the formula falls out.
The game show that fooled mathematicians
Three doors, one car, and a reason to switch.
The eight ways to move a square
The eight ways to turn or flip a square, and what happens when you do two in a row.
Four colours are always enough
Maps, crayons, and a proof done by computer.
Counting without double counting
Venn diagrams, and why nobody gets their own hat back.
The tower that takes forever
Recursion, and 585 billion years of moving disks.
The law of the first digit
Why 1 shows up far more than 9, and how it catches fraud.
A line, a curve and a rocket
Linear, quadratic, exponential, and who wins the race.
Arrows you can add
Vectors, rivers, and treasure maps.
The mind reader who survives a lie
Seven cards, one lie allowed, and the trick still works. Error-correcting codes.
Dice that go round in circles
You pick a die first. I can always pick one that beats it.
Hilbert's hotel: infinitely many buses
Infinitely many infinite buses, one halving trick, nobody left standing.
Nim where the last counter loses
Change one rule and find the exact spot where the perfect strategy breaks.
Add with a ruler
Three scales and a ruler that add any two numbers. Then solve an equation by drawing.
Negative numbers from pairs
Build the negative numbers from pairs of ordinary numbers, using only addition.
Tables where every equation has an answer
A Sudoku-style square used as a times table: one answer to every equation like a * x = c, yet the grouping rule breaks.
Connections music · astronomy · magic · nature · maps · sport · elections
More reading, fewer exercises. Where the math turns up when nobody told you it would.
Why there are twelve notes
Octaves are 2:1, fifths are 3:2, and the piano is slightly out of tune on purpose.
The size of everything
If the Sun is a basketball, where is Earth? And Kepler's law from a data table.
Math you can perform
Four tricks that always work, with the reasons they always work.
Why bees build hexagons
Tiling, perimeter, soap bubbles, and a storm on Saturn.
Finding a needle in twenty questions
Binary search, and why computers can find anything fast.
Why Greenland looks as big as Africa
You cannot flatten a sphere without cheating somewhere.
Why the best team doesn't always win
Series lengths, streaks, and luck in sport.
What day of the week were you born?
Leap years, and a trick to name the weekday of any date.
How an election can have no winner
Three friends, three dinners, and a paradox from 1785.
Why the 13th is most often a Friday
4,800 cases settle it forever. Friday wins by one.
Why can you balance a broom but not a pencil?
A marble in a bowl, a marble on a hilltop, and how a quick hand turns one into the other.
How far does a bouncing ball travel?
A model ball that bounces for ever, yet travels exactly 4 m.
Why bridges have gaps
A 100 m steel bridge grows about 1.2 mm for every degree it warms.
Model a cup of tea
A one-line rule for how tea cools, tested against thermometer readings.
The CT scan puzzle
Rebuild a hidden grid of numbers from the totals along its lines, the way a scanner does.
Your friends have more friends than you
On average, people's friends have more friends than they do. It is arithmetic, not bad luck.
Why does an ISBN sometimes end in X?
Old book numbers end in a check digit that makes a weighted sum a multiple of 11. Sometimes it has to be 10.
How far does a random walker get?
After 100 coin-flip steps left or right, a typical distance from the start is 10 steps, not 0 and not 100.
How steep can a sand pile be?
Dry sand piles up at about the same slope every time, so a big pile is a scaled-up small one.
Why is the shower too hot, then too cold?
You feel the water a few seconds late. A delay plus a strong reaction makes the temperature swing.
Why does a spoon get pulled into a tap stream?
A spoon is pulled into a stream of water, not pushed away. The same idea helps a wing fly, and the glide ratio says how well.
Why sunflowers count in Fibonacci numbers
Count a sunflower's spirals one way, then the other, and you often get neighbouring Fibonacci numbers. One simple rule for placing seeds does this.
Demos play with it
Every sheet that has an interactive version links to it. All the demos live in one place: the demo gallery.