Complete list of Puzzle articles

A Facebook friend puzzle
This puzzle is based on an actual situation I faced recently. I wanted to have a get-together with some coworkers from a previous job. The problem was that I was unsure if the guests all got along. I decided to do a simple test on Facebook. I personally was friends with ...

Monday puzzle: who will toss more heads?
Let's you and I play a game with coins. You have 100 dimes, and I have 99 pennies. At the same time, we will toss our coins in the air and let them fall on the floor. Then we meticulously count the outcomes of our tosses. You win if you show more ...

Monday puzzle: which lane is better?
You're on a two-lane highway and there are cars ahead of you in both lanes. You want to go straight ahead, but some of the cars in front of you might slow down to turn and delay you. You are in the left lane. In that lane, there are 3 cars, ...

Monday puzzle: the efficient drink order
Today's puzzle has a special extra. Because the problem is geometric in nature, I thought it would be fun to make a video of the puzzle on Youtube. Here is a link to the puzzle videos on Youtube. Efficient drink order problem - Youtube (it's just over at minute at 1 min 18 ...

Monday puzzle: pairs of cards
Let's play a game of chance. I have in my hand a standard 52 card deck. What I will do is flip over two cards at a time. If the two cards happen to be red, then you get the pair. If the two cards are black, then I get the pair. ...

Monday puzzle: Excel random numbers
The Excel function rand() generates a random number between 0 and 1. Let's say I generate two numbers using the rand() function. What is the probability the two numbers sum is less than 1 AND their product is less than 3/16? Give it a try, or experiment numerically, before reading the answer after ...

The three coin puzzle
After a lucky day in the casino, a gambler is greeted by a beautiful woman in the hotel lobby. She congratulates the gambler on his success, and wonders if he would like to parlay those winnings into even more money. She offers the following bet:

Monday puzzle: fruit label stickers
This is a problem slightly modified from a 2002 Putnam competition question. I thought about this problem while eating a large grapefruit one morning. The puzzle Imagine a perfectly round grapefruit that is labeled with 5 fruit stickers. Prove that the grapefruit can be cut in two equal halves in which one of the ...

Monday puzzle: random music
Have you ever felt your iPod or music player's shuffle function was not working correctly? Today's problem is about how random events are not always perceived as random.

Puzzle: birthday line problem
Birthday problems always make for fun probability puzzlers. There's the famous birthday problem, and then there is one I wrote about before called birthday holidays. The following is a puzzle I have adapted from the website braingle.

Monday puzzle: flip until heads
Probability calculations often show counter-intuitive results. Here is a puzzle about the distribution of coin flips.

Monday puzzle: the moat crossing problem
This is a fun geometrical puzzle--refer to the picture below. Here is the problem:

Puzzle: the Riddler vs Batman and Robin
This is a puzzle adapted from Max Schireson's blog. It is based on the trope that villains tend to use elaborate execution schemes. Here is the problem:

Monday puzzle: candy mixture problem
Today's puzzle is said to be based on an oddity noticed at a candy company. Imagine a company wants to make a new product that contains 4 different flavors of candy. The manufacturing process randomly fills the bags with each flavor. The company is worried that some bags might not ...

Monday puzzle: the train fly problem
This is a classic math puzzle. Here is the problem:

Monday puzzle: rope around the earth
This is a fun problem that first appeared in a 1702 book written by the philosopher William Whiston. Here is the puzzle:

Monday puzzle: digits in a book
I adapted this puzzle from a Wall Street Journal article about the new book Are you Smart Enough to Work at Google? I am quite excited about the book because it is written by William Poundstone, whose book I highly recommend. I have personally read How Would you Move Mount Fuji? ...

Monday puzzle: the three brick problem
This is a fun geometric problem with a practical solution: How can one measure the diagonal of a brick without any formula, using three bricks and a ruler? Can you solve it? Give it a try before reading the answer below.

Monday puzzle: lady tasting tea problem
In a previous post, I discussed ways to design a better blind taste test. This puzzle is about a taste test with historic significance to statistics.

Monday puzzle: coin flipping streaks
There are tons of probability puzzles related to coin flipping. This one is about the topic of hitting a lucky streak in flipping the same outcome repeatedly: (the puzzle is a problem from one of my college math books, Apostol Calculus Volume II)

A determinant game
This is a fun combinatorial game theory problem I came across. This problem appeared in the 2008 Putnam exam, an annual North American college math competition:

Answer to puzzle leaving work quickly
This post explains the solution to the puzzle posted on Monday called leaving work quickly. Okay, so sorry to keep changing the format of the Monday puzzle and solution. It's a work in progress so let me explain my logic. (And please forgive a broken link or two as I'm figuring ...

Monday puzzle: leaving work quickly
Alice and Bob were ready to leave the office when their mean boss assigned them more work. The boss told them to do the following boring things before they could go home: 1. Manually copy pages from bound books 2. Audit numbers in a spreadsheet 3. Fax documents to another office Each task takes 40 ...

Puzzle: ant and honey
The shortest distance between two points on a plane is a straight line. But finding the shortest distance on other surfaces is a more interesting problem. Here is a puzzle that is harder than it sounds:

A gift puzzle
With Christmas around the corner, I thought this puzzle would be appropriate. This is a nice problem that I found in a math puzzle book.

Auction theory puzzle: finding the right number of bidders
If you liked yesterday's puzzle about optimizing with uncertain demand, you'll definitely enjoy today's post. One of my favorite topics related to game theory is the subject of auction theory. The results from auction theory are very interesting, but I have yet to cover much about it because the math can be ...

Puzzle: how many Christmas trinkets to buy?
I came across a very interesting problem in the math section of reddit. The problem is about a business that wants to decide how much inventory of a product to buy facing an uncertain demand. The business wants enough to maximize profits, but it knows that unsold units will be ...

Cannibal game theory – a cool math puzzle
I came across a really interesting game theory problem at David Cowan's blog. The problem not only is about strategy, but its proof is interesting mathematically too. Here is the puzzle:

Puzzle: random size confetti
Today's puzzle is about statistical sampling, adapted from a problem I found in this book. Professor X teaches a probability class. He assigns a holiday-themed project to his students.

Alice and Bob play chess – a logic puzzle
Even if you don't like chess, you can appreciate today's puzzle. The problem does involve playing chess, but at its heart, the problem is a logic puzzle. Here is the setup:

Puzzle: a Christmas probability
This year Christmas will fall on a Sunday, but last year Christmas happened on a Saturday, and in general, the day can change from year to year. Because of the way dates rotate, you might think that Christmas can fall on every day of the week with equal probability. But in ...

Denny’s math commercial
I have a soft spot for TV commercials that mention math, as you may recall from my post about Little Caesar's pizza topping combinations. I recently came across a commercial for Denny's that also brings up a math problem involving combinations. It's a much easier problem but I still found it ...

A free throw game
In honor of the NBA lockout ending, today's post is related to basketball. It's a fun and relatively easy math problem about shooting free throws:

Puzzle: polynomial guessing game
If you love math as much as I do, then today's puzzle should be an absolute delight. The puzzle was originally submitted as "A Perplexing Polynomial Puzzle" in College Mathematics Journal, March 2005, p. 100. (link) H/T: Math Nexus, Delightful puzzles Here is the puzzle:

Puzzle: slicing up a pie
image by Paul Smith Holiday parties are one of the few times that I enjoy eating pies. This is a fun and easy problem about the mathematics of slicing a pie:

How many leftover menus can you make? (math problem)
After the feast of Thanksgiving, I enjoy slowly nibbling away at the leftovers. You'd be surprised how much variety you can get out of leftovers. Here's a math problem that illustrates the idea.

Puzzle: mischievous child
Kids can sometimes get out of control at parties, and today's puzzle is about a child that finds it amusing to mess with the drink setup. Here is the puzzle:

Puzzle: odds of a comeback victory
You're favorite sports team is down at halftime, but you are hoping they can pull it together and eek out a victory. What are the odds of that happening? The probability obviously depends on the exact game in question. But I came across an interesting probability problem that gives an interesting ...

Monday puzzle: the 6 glasses problem
Before I get to the puzzle, I have a small note about posts. Thanksgiving is a special time of year, and accordingly I am doing something special for this blog. This week Mind Your Decisions will run puzzles and fun problems all week long. I hope you enjoy them--regular posts ...

Company size vs talent: which one is more important? (Math problem)
There are always tradeoffs when picking one company over another. Imagine for a second you are thinking about picking between a standard position at a straight-forward, large company versus a fun position at an exciting smaller business. Which one is more appealing to you? There are many reasons people pick one ...

Monday puzzle: science experiment
This is a puzzle that my cousin sent to me that I really enjoyed: A chemistry teacher offers his class an experiment for extra credit. To complete the lab, students are to keep bacteria in a special chamber for exactly 9 minutes. The sadistic part is the teacher only gives the students ...

Ping pong probability
I came across a fun problem in this math book: Suppose A and B are equally strong ping pong players. Is it more likely that A will beat B in 3 out of 4 games, or in 5 out of 8 games? It's worth doing the math on this one. And once ...

Puzzle: the camel and bananas
This is a classic puzzle that I really enjoy: You want to transport 3,000 bananas across 1,000 kilometers. You have a camel that can carry 1,000 banana at most. However, the camel must eat 1 banana for each kilometer that it walks. What is the largest number of bananas that can be ...

Monday puzzle: two clock puzzles
With daylight savings ending this weekend, I thought it would be fun to include two time-related puzzles. They are challenging but not too difficult. Both puzzles involve old-fashioned analog clocks.

Monday Puzzle: Alice and Bob race to 1 million
This is a fun game in which Alice and Bob race to 1 million in a mathematical sense. Alice and Bob start with the number 1. Alice multiplies 1 by any whole number from 2 to 9. Bob then multiplies the result by any whole number from 2 to 9, and ...

A math game of dodgeball
Let's analyze a math game called dodgeball that's a sort of twist on tic-tac-toe. Here is how the game works. It's a two player game with the following set-up. Player 1 gets a 6x6 grid of squares as follows: Player 2 gets a 6x1 grid of squares: Here are the rules: 1. Player 1 begins ...

Monday puzzle: hide and seek with a twist
I came across this fun puzzle over at MathOverflow: A princess inhabits a flight of 17 rooms in a row. Each room has a door to the outside, and there is a door between adjacent rooms. The princess spends each day in a room that is adjacent to the room she ...

Puzzle: who wins this dart game?
Alice and Bob play the following game with their friend Charlie. Charlie begins the game by secretly picking a spot on the dartboard. The spot can be anywhere on the board, but once picked it does not change. Then Alice and Bob each get to throw one dart at the board. At this ...

Monday puzzle: train station pickup
This is a fun little puzzle about commuting from the train station. Mr. Smith, a commuter, is picked up each day at the train station at exactly 5 o'clock. One day he arrived at the train station unannounced at 4 o'clock and began to walk home. Eventually he met the chauffeur ...

A fun chess puzzle
Chess isn't something I normally write about, but this puzzle was too interesting to pass up. Why? This puzzle stumped some very smart people, including Ken Thompson (one of the people behind Unix and C), and even world class chess players like Garry Kasparov, Anatoly Karpov, and Mikhail Botvinnik, according to an ...

Monday puzzle: a really tough interview question
Many of my Monday puzzles have been technical interview brain teasers. Sebastian emails me an interview question that is a really difficult math problem. Hi Presh, This is a question I once got asked in an internship interview. The interviewer told me it was a hard problem, that would take at least 20 ...

Monday puzzle: elevator buttons malfunctioning (spoilers)
An elevator in my office building of 65 floors is malfunctioning. Whenever someone wants to go up, the elevator moves up by 8 floors if it can. If the elevator cannot move up by 8 floors, it stays in the same spot (if you are on floor 63 and press up, ...

Puzzle: how would you divide the land equally? (spoilers)
While I normally post math puzzles on Monday, I found another interesting puzzle so there is a bonus problem this week. Here is the problem. A father dies and wants to divide his land evenly amongst four sons. The plot of land has the following unusual shape: How can you divide the land ...

Probability question: would you play this card game?
Today I want to share a fun card game that is a twist on drawing to the high card. The game is played with a deck of cards numbered 1 to 100, and its rules are as follows. First, you and I each draw one card from the deck and show them ...

Monday puzzle: measuring ball bearings
This is an easy puzzle I came across. You are given a container that holds 24 ounces of ball bearings. You have a balance but no weights for the scale. You want to measure exactly 9 ounces. How can you do it? Give it a try before reading the answer below. image by winnifredxoxo

Monday puzzle: paying an employee in gold
In honor of Labor day, I thought I would offer the following interview brain teaser as today's puzzle. You have a solid gold bar, marked into 7 equal divisions as follows: | - | - | - | - | - | - | - | You need to pay an employee each ...

A fun number elimination game: who will win?
Bored at the airport, Alice and Bob decide to play the following mathematical game. Alice writes the numbers 1, 2, . . . , N on a piece of paper. Bob goes first, and he picks two numbers x and y from the list. Bob crosses out these numbers from the ...

Geometry puzzle: string cutting game
This is a great problem I came across, and I'd love to see it used as a consulting or technical interview question. Here is the setup. An interviewer gives you a string that measures 4 meters in length. The string is to be cut into two pieces. One piece is made into ...

I know how much money you are carrying – why the math trick works
I don't usually like generic email forwards, but I make an exception for ones that involve math tricks. A couple years ago I got an email about how a waiter could calculate your age using "restaurant math." It was an interesting trick and I posted about how the it worked: see ...

Puzzle: optimal guessing in a card game
You and a friend play a guessing game with cards. The deck consists of four black cards and four red cards. The game is played as followed. You announce a guess of a card, and then your friend flips over the card on the table to see if you are correct. The ...

Coin flipping game: how to make a fair toss from an unfair coin
Alice and Bob play a game as follows. Alice spins a coin on a table and waits for it to land on one side. If the result is heads, Alice wins $1 from Bob; if tails, Alice pays $1 to Bob. While the game sounds fair, Bob suspects the coin may be biased ...

Probability puzzle: what are the odds of a bad password?
The other day I got an email about an interesting probability problem: Hi Presh - I'm a regular reader of your site and I came across a question which has bugged me the last couple of days. The problem is as follows: A system has 100 accounts, two of which have bad ...

An easy birthday puzzle
image by y2bk Have you ever celebrated your birthday in another time zone? There's some novelty to it, as you can celebrate both during local time and your "home" time as you please. This sort of unusual time accounting can translate into a few extra hours of celebration. My friend loves doing this, ...

Puzzle: the math behind an Indian bowing custom
Growing up with an Indian heritage, I followed a tradition of bowing to elders as a sign of respect. On very special occasions, the custom was a bit more strict: each person in the group would have to bow once to each person who was older in age. For example: --in  a group ...

Puzzle: a probability drinking game
One night you find yourself drinking at a bar with a group of mathematicians. At the end of the night, the bar tab needs to be settled. To make things interesting, the group agrees to play the following game to determine who has to pay for everything. The game is played as ...

Math puzzle: chances of meeting up with a friend (and game theory puzzle extension)
Some nights that I go out to the bars it seems like I end up running into everyone. Other nights it's the exact opposite where I get out of sync and miss meeting my friends. The dynamics of meeting friends is rather interesting, and here is a probability puzzle that captures ...

Math puzzle: coin tossing carnival game
Over the holiday weekend, I went to a county fair and noticed a lot of carnival games. It got me thinking about a probability question arising from one of the games. Here is the puzzle: In a carnival game, you are to toss a coin on a table top marked with a ...

Puzzle: how often does it rain?
Weather is notoriously hard to predict. The following puzzle imagines a world where weather is simple and determined entirely by a set of probability rules. The puzzle: weather in mathland In Mathland, the weather is described either as sunny or rainy, nothing in between. On a sunny day, there is an equal chance ...

A fun graph theory puzzle
This is an interesting video I came across. It is very similar to the famous Seven Bridges of Konigserg problem. I especially like how the solution is explained in simple terms. The video is linked after the break: definitely worth a watch as it is just over 4 minutes long.

Birthday laws probability puzzle
Wouldn't it be fun if you could get a holiday every year on your birthday? Wouldn't it be even better if your friends got the same holiday too? Such a labor law would make for some fun parties. But it might have some interesting macro effects on the economy as a whole. Here's ...

Puzzle: escaping a thief
There's an interesting article at Fast Company that shows how games can help with strategic decision making. The US Navy is doing something very fascinating. It is launching an online multi-player game as a way to learn innovative tactics and strategies to combat Somali pirates. Time will tell if the ...

Video roulette math puzzle
I recently heard of a game called Netflix roulette where people pick a video at random and watch it regardless of the result. It got me thinking about probability and I came up with the following problem. Bob loves the TV show Law & Order. Each day he picks an episode at ...

Ants on a Triangle Puzzle
This is a fun little puzzle I was sent the other day. On a triangle, each corner has an ant. Each ant starts to move on an edge towards another corner, chosen at random. What is the chance that none of the ants collide? Can you solve it? I will post my ...

Math Problem: pizza topping combinations
One of my favorite commercials is an old ad for Little Caesar's pizza. They were offering a deal for ordering 2 pizzas, with up to 5 toppings on each. The commercial emphasized how customers could order pizzas in many different possibilities. The question is: exactly how many distinct ways are there to order ...

The world’s best tortilla puzzle
This is a brain teaser that I have adapted from the pizza puzzle found in the book The World's Best Puzzles. You start out with a whole, circular tortilla as pictured above. Your job is to divide the tortilla into eight equal pieces, using only cuts made in a straight line. What is ...

A math problem on Glenn Beck – can you solve it?
As I was flipping through the channels, I was surprised to see a math expression written on one of the chalkboards of the Glenn Beck program. And it wasn't something simple: the expression involved an infinite series. Can you solve the problem? (pictured below)

One mile south, one mile east, one mile north – a classic puzzle
I was having dinner with a friend the other day. We were talking about my writing, and out of the blue he asked me what my favorite puzzle was. A lot of the puzzles I wrote about on this site came to mind, including The hat puzzle A mystery Sherlock Holmes couldn't solve, ...

Broken sticks puzzle, and a seemingly paradoxical ratio
image via flickr, CC by 2.0 This is a neat little problem that is an extension of a puzzle from the Richard Wiseman's blog. The puzzle A warehouse contains thousands of sticks, each 1 meter long. One day a bored worker breaks each of the sticks in two, with each of the breaks ...

Math problem: passing the U.S. Citizenship test the lazy way
There's not much to prepare for in the U.S. citizenship test. Perhaps the hardest part is a section where they ask history and government questions. The interesting thing is how this section is structured. They ask a sample of questions of which one must answer a majority correctly. Here is how the ...

A gift giving puzzle (Secret Santa math)
Happy New Year! I am getting back in the groove of things, so posting might be slow for a couple weeks. In the meantime, another puzzle! During the holidays, I was part of a couple gift exchanges, and that experience reminded me about a problem that Chris mentioned a while ago ...

A fun prediction game
I came across a fun video made by Professor Richard Wiseman (who incidentally includes puzzles and optical illusions on his fantastic blog). It's amusing and somewhat related to decision trees. The video is after the break. Enjoy!

Puzzle: a drunkard and a cliff
This article is about a classic brain teaser that is tantalizingly easy to state but much harder to solve. I came across the puzzle in Frederick Mosteller's Fifty Challenging Problems in Probability with Solutions: From where he stands, one step toward the cliff would send the drunken man over the edge. He ...

Quick puzzle: how long to get to heaven?
For today, just a fun brain teaser that was passed along to me. A person dies and arrives at the gates to heaven. There are three identical doors: one of them leads to heaven, another leads to a 1-day stay in limbo, and then back to the gate, and the other ...

An interesting probability game
I came across a fascinating puzzle while perusing the book Game Theory Evolving. This problem is a bit more challenging mathematically, but I hope some of you will find it interesting. Here it is: Bob tells Alice to draw repeatedly from the uniform distribution on [0,1] until her current draw is less ...

Choosing the dealer in poker – is dealing to the first ace a fair system?
In Texas Holdem, sitting in the dealer position is a strategic advantage. The dealer position generally acts last in betting and is not forced to post blinds. For a game in progress, the dealer position rotates around the table after each hand. But at the start of the game, the dealer ...

A fun card puzzle
I came across a good brain teaser at this site: A pack of cards has 52 cards. You are in a dark room with this pack of cards. You have been told that inside the pack there are 42 cards facing down, 10 cards facing up. You have been asked to ...

A simple math puzzle about dice probability
I love reading about probability puzzles: even the easiest puzzles can take a moment to figure out exactly what is going on. The book Luck, Logic, and White Lies starts out with a fun little puzzle about dice throwing: With a pair of dice, one can throw the sum 10 either as ...

Why decision by committee often fails
Facing a tough choice, my boss set up a committee to decide. The committee spent days arguing and eventually came up with a horrendous recommendation. Sound familiar to you? image credit: dandechiaro I've wondered why committees often fail and make terrible decisions. Recently I came across a mathematical illustration that helps explain ...

Selling an extra ticket – a math bargaining puzzle
I came across a fun puzzle at a brain teaser forum BrainDen.com. Here is the problem: Sean wants to attend a basketball game. Unfortunately, he could not get tickets before it was sold out, but he has located someone that has an extra ticket which he is willing to sell ...

Guess the number – a game theory puzzle
I came across a highly amusing puzzle in Impossible?: Surprising Solutions to Counterintuitive Conundrums. What distinguishes this brain teaser from other "guess my number" puzzles is the method of solution. The answer is not an arithmetic trick or a mathematical sleight of hand like in your age by restaurant math. The puzzle ...

Game theory in Mario Party
Nintendo's Mario Party is a series that is a video game equivalent of board games. Gameplay involves mini-games which might include chance elements like rolling a dice or tactical elements like positioning a character on the right square. These video games consequently include many strategic elements and useful examples of ...

Game theory in Numb3rs: hide and seek
I occasionally enjoy watching the TV show Numb3rs. The show is about a team of FBI agents who solve crimes with the help of a fictional mathematics professor, Charlie Eppes. During a crime investigation, Charlie comes on the scene to analyze all of the numbers and look for patterns that ...

A math puzzle about splitting land
A father is splitting up land among his two sons in estate planning. How can he divide the land fairly? One approach is to split the land evenly. But even this method can get complicated if we add some realistic assumptions. Today's puzzle illustrates why splitting land can be a mind-boggling ...

Game theory and salary transparency
In 1994, shock jock Howard Stern created a public storm and ran for the governor of New York. His campaign was regarded as a publicity stunt, but some worried his radio popularity would convert into votes and make a mockery of American politics. Luckily, the fears never materialized. Stern dropped ...

16 fun applications of the pigeonhole principle
[This article is included in the 45th Carnival of Math] Mathematical logic can produce some great trivia. Did you know that at every instant, there is a spot in the world where no wind is blowing? It is true, and the proof comes as an application of a fixed point theorem ...

The Dice Brain Teaser: A Technical Interview Question that Can Help You Solve Problems Better
Today's puzzle is special. I like it even more than previous brain teasers I've discussed, like the monk problem and the hat puzzle. Like the others, this is a technical question sometimes asked during job interviews (hat tip: Reasonable Deviations). But unlike the others, this problem serves as an introduction to some ...

The Hat Puzzle (A Consulting or Engineering Interview Brain Teaser)
The Hat Puzzle is one of my favorites. This is one of those brain teasers people might ask in an interview for a consulting or engineering job. Here is the problem, as worded by Sara Robinson (this is a good article to read): Three players enter a room and a red or ...