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Math, Going Further, Grades 9–12

Puzzles past the course for students in Grades 9 to 12: each starts with a question to try on the page, then shows the mathematics that answers it, and how far it goes. Not in any exam.

Start reading: Topic 1 · Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? →

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Growth, logarithms and scales

  1. Topic 1Logarithms and log scales Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? One step on the earthquake scale means about 32 times the energy. Why measure the world this way?  
  2. Topic 2Logarithms and the slide rule How Did a Book of Numbers Make Multiplying Easy? For 350 years scientists multiplied by adding, with a book of tables or a sliding ruler. How?  
  3. Topic 3Continuous growth and e What Is e, and Why Does Growth Keep Finding It? Interest every year, every month, every second: does the money grow without end? It stops at e.  
  4. Topic 4Ratios and logarithms in music Why Does a Scale Have Twelve Notes? Notes that sound sweet together have simple ratios. Twelve steps nearly fit them all, but never quite.  
  5. Topic 5Very large numbers What Is the Biggest Number Anyone Has Used? Archimedes counted the grains of sand that would fill the universe. Mathematicians have needed far bigger.  

Whole-number puzzles

  1. Topic 6Diophantine equations Can You Buy 100 Birds with 100 Coins? A puzzle from 1,500 years ago: only whole birds, only whole coins. How many answers, and how do you find them?  
  2. Topic 7Pythagorean triples How Do You Find Every Right Triangle with Whole Sides? 3, 4, 5 is a right triangle. So is 20, 21, 29. Is there a recipe that makes them all?  
  3. Topic 8The Chinese remainder theorem How Many Eggs Were in the Basket? Count the eggs in twos, threes, fours, fives, sixes: one is always left over. How many were there?  
  4. Topic 9Modular arithmetic and public-key codes How Can Two Strangers Agree on a Secret in Public? Every message your phone sends is locked with a key nobody sent. Remainders make it possible.  
  5. Topic 10Unsolved problems What Don’t Mathematicians Know About Prime Numbers? Every even number seems to be two primes added. Nobody has proved it. Simple questions, still open.  

Mental math

  1. Topic 41Calculation tricks, their algebra and their history Why Do the ‘Vedic Mathematics’ Tricks Work? Multiply in one line, subtract from 1,000 at a glance. Why do the tricks work, when are they slower, and did they really come from the Vedas?  

Counting and chance

  1. Topic 11The pigeonhole principle Must Two People Here Share a Birthday Month? Thirteen people, twelve months: two must match. A simple idea that proves surprising things.  
  2. Topic 12The birthday problem Will Two People in Your Class Share a Birthday? With 23 people, it is more likely than not. Most people guess far more. Where does intuition go wrong?  
  3. Topic 13Conditional probability Should You Switch Doors? A prize behind one of three doors, a host who knows, and a choice that fooled professors. Switch or stay?  
  4. Topic 14Base rates and Bayes’ rule Is a Positive Test Result Bad News? A test is 99% accurate and it says yes. The chance you have the disease may still be small. Why?  
  5. Topic 15Expected value What Is a Game Worth on Average? A coin game whose average prize is endless, yet nobody would pay much to play. What is it worth?  
  6. Topic 16Recurrences and counting How Many Ways Can You Climb the Stairs? One step or two at a time: how many ways up ten stairs? Count small cases and a famous pattern appears.  

Logic and proof

  1. Topic 17Proof by contradiction Can You Prove √2 Is Not a Fraction? Calculators show √2 as 1.414213…, never ending. How can anyone be sure no fraction is exactly it?  
  2. Topic 18False proofs in algebra What’s Wrong with This Proof That 1 = 2? Every line looks right, yet the end is absurd. Finding the broken step teaches what algebra really allows.  
  3. Topic 19Paradoxes of self-reference Can a Sentence Say It Is False? “This sentence is false.” If it is true, it is false. Puzzles like this shook the foundations of maths.  

Shapes and space

  1. Topic 20Geometry on a sphere Why Do Flights to America Fly Near the North Pole? On a globe the straightest route looks curved on a map, and a triangle can have three right angles.  
  2. Topic 21Fractals and the coastline paradox How Long Is Nepal’s Border? Measure it with a longer ruler and it gets shorter. A border may have no true length at all.  
  3. Topic 22Aperiodic tilings Can a Pattern of Tiles Never Repeat? In 2023 an amateur found one tile that covers a floor forever with no repeating pattern. How can that be?  
  4. Topic 23The golden ratio: facts and myths Is the Golden Ratio Really Everywhere? Shells, faces, temples and paintings are all said to hide 1.618. Which claims survive a ruler?  
  5. Topic 24Platonic solids Why Are There Only Five Perfect Dice? Fair dice with identical faces come in only five shapes. A short argument shows why there can’t be a sixth.  
  6. Topic 25Impossible constructions Can You Square a Circle? With ruler and compass, Greeks tried for centuries to make a square equal to a circle. It was proved impossible.  

Networks and routes

  1. Topic 26Minimum spanning trees How Do You Connect Every Village with the Least Cable? Link every village to the grid with the least wire. A greedy rule turns out to be exactly right.  
  2. Topic 27Shortest paths How Does a Map App Find the Quickest Route? Millions of roads, an answer in a thousandth of a second. The trick: aim the search, and do the work before anyone asks.  
  3. Topic 28Round robins and coloring How Do You Schedule a League So Everyone Plays Everyone? Ten teams, every pair plays once, nobody plays twice in a round. Is there always a timetable?  

Data and reasoning

  1. Topic 29Simpson’s paradox Can a Treatment Win in Every Group but Lose Overall? Better for the young, better for the old, worse for everyone together. Real data does this.  
  2. Topic 30Correlation and cause Does Ice Cream Cause Drowning? Two things rise together. Does one cause the other, or is something else behind both?  
  3. Topic 31Sampling and margin of error Can 1,000 People Speak for 3 Crore? A survey asks a thousand people and claims to know the country. When is that fair, and how sure can it be?  
  4. Topic 32Benford’s law Why Do So Many Numbers Start with 1? In real data, about 30% of numbers begin with 1 and under 5% with 9. Auditors use it to catch fraud.  

Modelling the real world

  1. Topic 33Models of growth When Will Nepal’s Population Stop Growing? Nepal’s censuses since 1911 trace a curve. Which formula fits it, and what does each one predict?  
  2. Topic 34Estimating with real data How Much Rain Falls on the Kathmandu Valley? A few millimetres on a gauge, an area on a map: turn them into litres, tankers and lakes.  
  3. Topic 35The mathematics of voting Which Voting System Is Fair? The same votes, counted three fair-sounding ways, can elect three different winners.  

Old problem books

  1. Topic 36Pell’s equation and the chakravala How Did Bhaskara Solve Fermat’s Challenge 500 Years Early? In 1657 Fermat dared Europe to solve x² = 61y² + 1. Bhaskara had done it in 1150.  
  2. Topic 37Infinite series for π How Did Madhava Find π Without a Calculator? In Kerala, 600 years ago, mathematicians added endless fractions to find π, centuries before Europe.  

Infinity and paradoxes

  1. Topic 38Countable and uncountable sets Are Some Infinities Bigger Than Others? There are as many even numbers as whole numbers, but more points on a line. A clever list proves it.  

Decisions and games

  1. Topic 39Game theory and the prisoner’s dilemma Should You Cooperate or Cheat? Two players, each better off cheating, both worse off when they do. How does cooperation survive?  
  2. Topic 40Fair division How Do You Share an Inheritance Fairly? Land, a house and a buffalo, three heirs who value them differently. Can everyone feel they got the most?  
  3. Topic 42Compound returns, random walks, survivorship bias and diversification Can Anyone Beat the Stock Market? Down 50%, then up 50%: are you back where you started? The maths behind share prices, tips, charts and fees.  

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.