Chapter 5: Number Play

The whole numbers $1, 2, 3, 4, \dots$ look ordinary at first glance. Once you start playing with them, they turn out to be full of surprises , secret shortcuts that tell you instantly whether a number is divisible by $11$, magic patterns where every row and column adds to the same total, and codes built on nothing but remainders.

This chapter is about playing with numbers. You will pick up a small toolkit of divisibility tests that work without long division. You will meet modular arithmetic , the way of thinking about remainders that runs everything from calendars to credit-card check digits. You will also peek at famous puzzles such as magic squares, cyclic numbers, and digit-sum tricks that astonish friends.

The real reason for all this play is that it builds mathematical intuition. The patterns you find here will help you spot factorisations, simplify fractions in your head, and check your arithmetic for blunders. Mathematicians sometimes say that "playing with numbers" is mathematics; everything else is just writing it down.

By the end you will be able to test divisibility by $2, 3, 4, 5, 6, 8, 9, 10, 11$ at a glance, use remainders to solve clock-style problems, and explain why certain digit tricks always work.

What's inside

1. Divisibility tests , quick checks for $2, 3, 4, 5, 6, 8, 9, 10, 11$.
2. Remainders and modular thinking , clocks, calendars, and check digits.
3. Digit puzzles , magic squares, palindromes, and cyclic numbers.
4. Number sequences and patterns , arithmetic, geometric, Fibonacci.
5. Cryptarithms and challenges , solve letter-for-digit puzzles.

Key results

Modular notation. "$a \equiv b \pmod{m}$" means $a$ and $b$ leave the same remainder when divided by $m$. For example, $17 \equiv 5 \pmod{12}$ because both leave remainder $5$ when divided by $12$.

How to read this chapter

Keep a pencil and paper. Try every test on a few numbers of your own before reading the next one. The puzzles in the last subtopic are the best part , try them with a friend.