Open problems — chapters of the maths textbook Queen of the Sciences

0 What mathematics is Five points on a circle cut it into 16 pieces; six points give 31, although your hand wants to write 32. The course begins with this trap: why examples are never enough in mathematics, what replaces them, and which seven questions lie ahead. Ages 11–15 30 min

1 Counting: from notches to bits How do you write a number too big for your fingers or for notches on a stick? A walk through the halls of a museum of counting, from a notched bone to binary code, to see how people learned to pack any number into a few signs. Ages 11–15 40 min

2 Zero and minus What is 3 − 5? A seventeenth-century sceptic is sure it's nothing, and he calls “minus times minus” nonsense. Let's argue with him by the rules: every sign rule will have to be defended. Ages 11–15 40 min

3 Prime numbers Some numbers can't be broken into factors. Everything else is built from them by multiplication, there are infinitely many of them, and nobody can say in advance where the next one will turn up. Ages 11–15 40 min

4 Divisibility and Euclid's algorithm Puzzles about a calendar, a long number, tiles, jugs and coins, and one answer to all of them: the greatest common divisor, and a way to find it that is more than two thousand years old. Ages 11–15 40 min

5 Fractions How to share seven pies fairly among three people, why you flip the fraction when you divide, and where the endless merry-go-round of the digits 142857 in one seventh comes from. Ages 11–15 40 min

6 Root two and the real numbers The case of the square's diagonal: three independent expert reports prove that its length can't be written as a fraction. We look at how the Babylonians lived with it, what continued fractions are, and where the real numbers come from. Ages 11–15 45 min

7 Letters for numbers A magician guesses the number you thought of and does mental arithmetic faster than a calculator. We'll expose his tricks one by one, and along the way learn to write down a rule that holds for all numbers at once. Ages 11–15 40 min

8 Equations: the art of balance An equation is a pair of scales in balance with a box of unknown weight on it. We'll learn to take things off the pans without making the scales tip, and find out where the word “algebra” itself comes from. Ages 11–15 40 min

9 Functions and graphs Sometimes the answer to a question isn't a number but a rule. Let's play with machines that turn some numbers into others: learn to crack them from a few throws, see them whole in a single picture, tune them, chain them into a production line and run them in reverse. Ages 11–15 40 min

10 Quadratic equations The problem the last chapter ended with is nearly four thousand years old. Let's excavate it layer by layer, from a clay tablet to the quadratic formula, and see what the formula really says. Ages 11–15 40 min

11 Systems of equations and inequalities When there are several conditions, one equation isn't enough. Let's open a small toy workshop: find prices from two invoices, work an ancient Chinese counting board, run into the limits of the storeroom and find out why the most profitable plan always sits in a corner. Ages 11–15 45 min

12 Powers and logarithms A quantity that is multiplied at every step sooner or later outgrows any quantity that is added to, and usually sooner than you'd think. Let's hold a race of growth rates, then find a tool that turns multiplication into addition and tames astronomical numbers. Ages 11–15 40 min

13 Sequences and induction How to add a hundred numbers in a minute, why an infinite sum can be finite, and how to prove a statement for every n at once: by standing dominoes in a row and pushing the first one. Ages 11–15 45 min