Presh Talwalkar - Math Puzzles Volume 2: More Riddles And Brain Teasers In Counting, Geometry, Probability, And Game Theory

Math Puzzles Volume 2: More Riddles And Brain Teasers In Counting, Geometry, Probability, And Game Theory

About The Author

Presh Talwalkar studied Economics and Mathematics at Stanford University. His site Mind Your Decisions has blog posts and original videos about math that have been viewed millions of times.

Books By Presh Talwalkar

The Joy of Game Theory: An Introduction to Strategic Thinking . Game Theory is the study of interactive decision-making, situations where the choice of each person influences the outcome for the group. This book is an innovative approach to game theory that explains strategic games and shows how you can make better decisions by changing the game.

Math Puzzles Volume 1: Classic Riddles And Brain Teasers In Counting, Geometry, Probability, And Game Theory . This book contains 70 interesting brain-teasers.

Math Puzzles Volume 2: More Riddles And Brain Teasers In Counting, Geometry, Probability, And Game Theory . This is a follow-up puzzle book with more delightful problems.

Math Puzzles Volume 3: Even More Riddles And Brain Teasers In Geometry, Logic, Number Theory, And Probability . This is the third in the series with 70 more problems.

But I only got the soup! This fun book discusses the mathematics of splitting the bill fairly.

40 Paradoxes in Logic, Probability, and Game Theory . Is it ever logically correct to ask May I disturb you? How can a football team be ranked 6th or worse in several polls, but end up as 5th overall when the polls are averaged? These are a few of the thought-provoking paradoxes covered in the book.

Multiply By Lines . It is possible to multiply large numbers simply by drawing lines and counting intersections. Some people call it how the Japanese multiply or Chinese stick multiplication. This book is a reference guide for how to do the method and why it works.

The Best Mental Math Tricks . Can you multiply 97 by 96 in your head? Or can you figure out the day of the week when you are given a date? This book is a collection of methods that will help you solve math problems in your head and make you look like a genius.

Why Math Puzzles?

I am adding this introductory note in 2015 after completing volume 3 of the math puzzles series. What is the point of all of these math problems?

From a practical perspective, math puzzles can help you get a job. They have been asked during interviews at Google, Goldman Sachs, as well as other tech companies, investment banks, and consulting firms.

Math puzzles also serve a role in education. Because puzzles illustrate unexpected solutions and can be solved using different methods, they help students develop problem solving skills and demonstrate how geometry, probability, algebra, and other topics are intimately related. Math puzzles are also great for practice once you are out of school.

But mostly, math puzzles are worthwhile because they are just fun. I like to share these problems during parties and holidays. Even people who do not like math admit to enjoying them. So with that, I hope you will enjoy working through this collection of puzzles as much as I have enjoyed preparing the puzzles and their solutions.

Each puzzle is immediately accompanied with its solution. I have never been a fan of how print books put all the solutions at the end--it is too easy to peek at the solution for another problem's solution by mistake. In any case, while you are working on a problem, avoid reading the following section which contains the solution.

This book is organized into topics of counting, geometry, and probability and game theory. In each section, the puzzles are roughly organized with increasing difficulty. It is never easy to organize puzzles by difficulty: some of the hard puzzles may be easy for you to solve and vice versa. But as a whole, the harder puzzles tend to involve higher-level mathematics, like knowledge of probability distributions or calculus.

Table Of Contents

Section I: Counting

These problems are generally about the number of ways to do things, but there are a few algebra and logic problems included as well.

Puzzle 1: 3 Lamps

In front of you are 3 switches, each of which operates a lamp in another room not visible to you. Your job is to identify which switch operates which lamp.

You are allowed to operate the switches, but you can only visit the other room once. How can you identify which switch operates which lamp?

My friend was asked this problem in an interview for a consulting job. It requires logical thinking and applying a bit of common sense.

Answer To Puzzle 1: 3 Lamps

Turn on one of the switches for a few minutes. Then turn it off and turn on another switch.

When you visit the other room, one lamp will be on. That corresponds to the last switch turned on.

Then touch the other two lamps and feel which bulb is warm. That corresponds to the switch you had turned on for a while and then turned off. The last lamp corresponds to the switch you did not touch at all.

Puzzle 2: Famous Logic Puzzle

This problem was asked in a famous psychological experiment.

Four cards are on a table and each has a number written on one side and a color on the other side. The faces of the cards showing are 3, 8, red, and brown. You are told that every card that shows an even number on one side has the color red on the other side.

Which cards do you need to flip over to test this rule?

Only about 10 percent of subjects got the answer correct. Can you figure it out?

Answer To Puzzle 2: Famous Logic Puzzle

The cards that need to be flipped over are "8" and "brown". The rule is only broken if an even-card has another color, or a card with another color has an even number on the other side. So the only cards to check are "8" and "brown."

Here is the logic in more detail when deciding whether each card should be flipped.

Flip over 3? The rule states the opposite side of even cards has red. The rule does not say what the opposite of an odd card has. We could have cards with an odd number on one side and red on the other. There is no need to flip 3.

Flip over 8? If the rule is true, this card better have red on the other side so "8" should be flipped.

Flip over red? The rule states the opposite side of even cards has red. It does not say what the opposite side of red cards has. So if we flip over the red card to find an odd number, that does not break the rule. No need to flip it.

Flip over brown? If we flip this card over and find an even number, then that would break the rule, because it would be an even card that has brown on the other side. Hence we must flip this card over.

The puzzle is not very hard, but the situation appears to make the logic hard to follow.

Experiments have found re-phrasing the problem in a familiar context is helpful. For instance, consider the rule that every voter must be over 18 years old. You are given profile cards with age on one side and whether a person voted on the other. The cards shown to you are, "16", "voted", "23", and "not voted". Which cards do you need to flip over to make sure everyone who voted was legally over 18 years old?

People generally solve this problem instantly (you should check "16" and "voted").

This problem is known as the Wason Selection Task.

Puzzle 3: Lost Money To A Buyer

I sold a concert ticket for $20 to someone responding to a Craigslist ad. The buyer paid with a $50 bill.

I didnt have change on hand, but luckily my friend did. I gave him the $50 note for smaller bills, and I provided the buyer with $30 in change.

The next day my friend tried to use the bill, but it was counterfeit and not accepted.

The buyer did not respond to my emails and I realized I was probably scammed. I owned up to the mistake and repaid my friend $50.