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Math, Going Further, Grades 6–8

Puzzles past the course for students in Grades 6 to 8: each starts with a question to try on the page, then shows the mathematics that answers it. Not in any exam.

Start reading: Topic 1 · How Much Rice Is on the Chessboard? →

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Big numbers and growth

  1. Topic 1Exponential growth and large numbers How Much Rice Is on the Chessboard? One grain on the first square, then double on every square. How could a king run out of rice?  
  2. Topic 2The size of big numbers How Long Is a Crore Seconds? You can say “a crore” and “a billion” easily. Can you feel how big they are, and how different?  
  3. Topic 40Growing dominoes: geometric growth and energy How Big a Domino Can a Small Domino Knock Over? A falling domino can topple one about 1.5 times its size. Start from a rice-grain domino: how many to topple one taller than your school?  

Whole-number puzzles

  1. Topic 3Number tricks and algebra Why Does 1089 Always Come Back? Pick any three-digit number, reverse, subtract, reverse, add. Why does everyone end at the same answer?  
  2. Topic 4Place value and remainders Why Do the Divisibility Tests Work? Add up the digits and you know whether 9 divides the number. Why should the digits know?  
  3. Topic 5Primes and Euclid’s proof Is There a Biggest Prime Number? Primes get rarer as numbers grow. Do they ever stop? A proof from 2,300 years ago says no.  
  4. Topic 6Clock arithmetic What Day of the Week Will Your 100th Birthday Be? Today is a Friday. What day will it be in 1,000 days? You don’t need a calendar, only remainders.  
  5. Topic 7Ciphers and remainders Can You Write a Message Only a Friend Can Read? Shift every letter three places and a note turns to nonsense. How easily can a stranger crack it?  
  6. Topic 8The coin problem Which Amounts Can You Never Pay with 3s and 5s? With only Rs 3 and Rs 5 stamps, some amounts are impossible. Find the biggest, and see why it stops.  
  7. Topic 9Magic squares How Does a Magic Square Work? Every row, column and diagonal adds to the same number, carved in a temple nearly a thousand years ago.  

Mental math

  1. Topic 37Mental multiplication and algebra How Can You Multiply 98 × 97 in Your Head? Some people answer 98 × 97 before you have found a pencil. Cut up a square of 100 by 100 and you can too, then invent a trick of your own.  
  2. Topic 38Estimation and casting out nines Is the Answer Roughly Right? A calculator is never wrong, but the finger typing into it can be. How can you tell in seconds that an answer is off?  
  3. Topic 39The abacus and place value How Does an Abacus Think? One bead worth five and four worth one on each rod. How can children who learn it add in their heads by picturing the beads?  

Counting and chance

  1. Topic 10Counting by multiplying How Many Number Plates Can a Province Make? Letters and digits on a plate: how many cars before the numbers run out? Count without listing.  
  2. Topic 11Counting pairs How Many Handshakes at a Party of 30? Everyone shakes hands with everyone once. Is it 30 × 30? Count it three ways and they all agree.  
  3. Topic 12Paths and Pascal’s triangle How Many Ways Can You Walk to School? Only east and north along the lanes: how many different routes? A triangle of numbers knows.  
  4. Topic 13Outcomes and probability Which Total Comes Up Most with Two Dice? Roll two dice and add. Is 12 as likely as 7? Roll a hundred times, then find out why.  
  5. Topic 14Fair games and expected value Is This Game Fair? A fairground game looks generous. Play it a thousand times on paper and see who really wins.  

Logic and proof

  1. Topic 15Logic puzzles Who Is Telling the Truth? On an island, some people always lie and the rest always tell the truth. Two words can unmask them.  
  2. Topic 16Patterns and counterexamples Is a Pattern That Works Ten Times Always True? 1, 2, 4, 8, 16… the next is surely 32. Draw it and count: it is 31. When can you trust a pattern?  
  3. Topic 17False proofs in pictures Where Did the Missing Square Go? Move four pieces around and a hole appears from nowhere. Something in the picture is lying.  

Shapes and space

  1. Topic 18Proofs of Pythagoras’ theorem How Many Ways Can You Prove Pythagoras’ Theorem? You checked it with cut-out squares. People have proved it hundreds of ways, one, the story goes, in a single word: “Behold!”  
  2. Topic 19Shortest paths and reflection What Is the Shortest Way to the River and Back? Walk from home to the river for water, then to the field. Where on the bank should you stop?  
  3. Topic 20Fibonacci numbers Why Do Sunflowers Count in Fibonacci Numbers? Count the spirals on a sunflower head and you get 34 and 55. Why would a flower know these numbers?  
  4. Topic 21Fractals Can a Shape Have an Edge That Never Ends? A snowflake whose edge grows longer each time you look closer, yet fits on a page. How?  
  5. Topic 22Angles and proportion How Did Eratosthenes Measure the Earth? Two shadows in two cities, 2,200 years ago, gave the size of the whole Earth. You can do it too.  
  6. Topic 23Euler’s formula for solids Why Do Solids Always Obey One Rule? Count the corners, edges and faces of any solid you like. One simple sum always gives 2. Why?  

Networks and routes

  1. Topic 24Euler paths Can You Cross Every Bridge Just Once? A city with seven bridges, and a walk that crosses each one exactly once. Can it be done?  
  2. Topic 25Planar networks Can Three Houses Reach Three Wells Without Paths Crossing? Three houses, three wells, a path from every house to every well, and no paths may cross. Try it.  
  3. Topic 26Shortest routes What Is the Quickest Way Across the Map? Many roads, each with its time. Is there a way to find the fastest route without trying them all?  

Data and reasoning

  1. Topic 27Misleading graphs Can a Graph Lie? Two graphs of the same numbers: one shows a crisis, one shows nothing. Learn the tricks.  
  2. Topic 28Mean, median and outliers What Does an Average Hide? One rich person walks into a tea shop and the average income soars. Who got richer?  
  3. Topic 29Estimating with Fermi questions How Much Rice Does Nepal Eat in a Day? Nobody counts it, but you can work it out to within a factor of two with what you already know.  

Old problem books

  1. Topic 30Bhaskara’s verse problems Can You Solve the Puzzles of the Lilavati? A swarm of bees, a flock of geese, a peacock and a snake: maths set as poems nearly 900 years ago.  
  2. Topic 31Old ways to multiply How Did People Multiply Before Calculators? Doubling and halving, lattices, rods of bone: clever ways to multiply that still work today.  
  3. Topic 32Zero and place value Where Did Our Zero Come From? Ten digits can write any number, thanks to a small circle that means nothing. Who first wrote it?  

Infinity and paradoxes

  1. Topic 33Infinite sums Can You Add Halves Forever and Still Get One? Walk half way to the door, then half of what is left, forever. Do you ever get there?  
  2. Topic 34Infinite sets Is There Room at a Full Hotel with Endless Rooms? Every room is taken, a guest arrives, and the manager still finds a bed. Infinity plays by its own rules.  

Games and strategy

  1. Topic 35Winning positions Who Wins the Race to 20? Take turns adding 1 or 2; whoever says 20 wins. One player can always win. Which, and how?  
  2. Topic 36Fair division How Do You Cut a Cake Fairly for Three? For two it’s easy: one cuts, the other chooses. For three people nobody may feel cheated. How?  

Practice

How to use this book

These topics go past the school course, for anyone who wants to know more. Each one starts from things you learned in class, linked under its title, so you can look back. Each is a question, with the subject it belongs to beside its number. They are not in any exam. Inside you’ll find:

  • Worked examples: try them first; the answer stays hidden until you open it.
  • Try it questions and each topic’s questions at the end, with hints and answers to open when you’re ready.
  • Diagrams to play with: drag, slide and change things to see what happens.
  • Did you know? boxes with the stories and surprises behind the maths.

Every fact here has been checked against reliable sources, but these topics have not yet been reviewed by subject experts. If you find a mistake, please tell us.