This is a collection of my favorite questions, typically classified as
*recreational mathematics* or *logic puzzles*, that
engage students
in a way the uninspiring curriculum in a usual math class does not.
Email me if you know the source of any of these I’ve left unattributed.
Also see my
links to similar collections at the bottom of this page.

### Party of Six

There are six people at a party. Every two people at the party are either mutually

good friendsor mutuallytotal strangers; there is no in-between, no acquaintances. Prove that there must be at least one trio of people at the party who are all mutually good friends with each other,orare all mutually total strangers to each other.

### The Flight Around the World

Imagine a planet covered entirely with water with the exception of a single small island. On this island is a sizeable fleet of airplanes and an endless supply of fuel. These airplanes are very peculiar though: they can carry only enough fuel to fly half-way around the planet, and they are capable of exchanging fuel between themselves instantaneously midair. What is the fewest number of planes that would have to be used to allow a single plane to fly completely around the planet and back to the small island along a great circle?

### Forehead Dots

Three friends Anita, Becca, and Charleston are challenged to a game by the Game Maestra. The Game Maestra paints two colored dots on each of the friends’ foreheads and tells the friends that each dot is either blue or yellow, but neither color is used more than four times. She then places the three friends in a circle so that each of them can see the dots on their friends’ foreheads, but not the dots on their own. The game proceeds like this: The Maestra will ask the friends in turn, first Anita, then Becca, then Charleston, then Anita again, then Becca again, and so on, if they know the colors of the dots on their foreheads. When someone responds “no,” the Maestra asks the next person. If someone responds “yes” and is right, the friends win! However if someone responds “yes” and is wrong, all three friends will be banished to the shadow realm for eternity.

The friends were given no time to strategize, but they begin playing. Their responses in turn are “no,” “no,” “no,” “no,” “yes,” and the three friends win! Whare are the colors of the dots on Becca’s forehead?

### The Monkey and the Coconuts

Five women and a monkey were shipwrecked on a desert island. They spent the first day gathering coconuts for food, piling up all the coconuts together before they went to sleep for the night. But while they were all asleep, one woman woke up and thought there might be a row about dividing the coconuts in the morning, so she decided to get up and take her share now. She divided the coconuts into five equal piles except for one coconut left over which she gave to the monkey, and she hid her pile and put the rest back together. By and by, another woman woke up and did the same thing, and similarly she had one coconut left over which she gave to the monkey. And each of the five women did the same thing, one after the other; each one taking a fifth of the coconuts in the pile when she woke up, and each one having one left over for the monkey. In the morning they divided what coconuts were left, and they came out in five equal shares. Of course each one must have known that there were coconuts missing, but each one was as guilty as the others, so they didn’t say anything. How many coconuts were there in the beginning?

### Einstein’s Puzzle

There are five houses of different colors next to each other on the same road. In each house lives a woman of a different nationality. Each woman has her favorite drink, her favorite brand of cigarettes, and keeps a particular kind of pet.

The Englishwoman lives in the red house.

The Swede keeps dogs.

The Dane drinks tea.

The green house is just to the left of the white one.

The owner of the green house drinks coffee.

The Pall Mall smoker keeps birds.

The owner of the yellow house smokes Dunhills.

The woman in the center house drinks milk.

The Norwegian lives in the first house.

The Blend smoker has a neighbor who keeps cats.

The woman who smokes Blue Masters drinks bier.

The woman who keeps horses lives next to the Dunhill smoker.

The German smokes Prince.

The Norwegian lives next to the blue house.

The Blend smoker has a neighbor who drinks water.Who keeps fish?

### Blue-Eyed Villagers

On a remote mountaintop there’s a small village of one hundred people. Everyone in the village has either brown eyes or blue eyes, and devoutly follows the same religion that has a single peculiar feature: each villager is forbidden from knowing the color of their own eyes, or even from discussing the topic. So each villager sees the eye color of every other villager, but has no way of discovering their own. (They’ve forbidden reflective surfaces long ago.) If a villager does discover their eye color, their religion compels them to proceed to the village square at high noon the following day, and call out for all their fellow villagers to come witness as they pluck out their eyes.

Of the one hundred villagers, it turns out that ten of them have blue eyes and the rest have brown eyes, although they are not aware of this fact. Each villager is highly logical and religiously devout, and they all know that each villager is highly logical and devout (and they all know that they all know that each villager is highly logical and religiously devout, and …).

One day a blue-eyed mountaineer, tired and hungry, stumbles across the village asking for help. That evening at the communal village dinner she addresses the whole gathering to thank them for their hospitality. However, not knowing their customs, the mountaineer remarks in her address

how unusual it is to see another blue-eyed person like myself in this remote region. What effect, if any, does this faux pas have on the village?

### Weighing Coins

The Game Maestra presents you with nine gold coins and tells you that exactly one of them is counterfeit, and weighs slightly less than the other eight. She gives you a balance scale with the coins and challenges you: “Identify the counterfeit coin using the balance scale only twice.”

Once you answer this challenge though the Game Maestra bursts out “Ha! That was simply a warm up.” She takes those nine coins, throws them out the window, and pulls twelve silver coins from her pocket and challenges you: “Identify the counterfeit coin among these using the balance scale only three times. And this time, I want

youto tellmewhether the counterfeit coin is heavier or lighter than the other eleven.”

### Quarters on a Table

Aldous and Buxley sit together at a circular table in a laundromat, bored, waiting for their clothes to dry. Fiddling with their hoard of laundry quarters, they create a game. They take turns placing quarters flat on the table. Once a quarter is placed, it cannot be moved. Quarters’ edges can touch, and they can hang over the edge of the table, but may not lay on each other: they

mustlay flat on the table. The last person who can place a quarter on the table is the winner. Prove that, if he plays cleverly, whoever goes first can always win.

### Triangles in the Colored Plane

Suppose that in the plane (or in \(\mathbf{R}^2\) if you prefer to call it that) every point is colored either red or blue. Show that no matter how the points are colored, there has to exist an equilateral triangle somewhere in the plane such that the vertices of the triangle are all the same color.

### Two Brothers Decide to Run a Race

Two brothers decided to run a \(100\)-meter race. The older brother won by \(3\) meters. In other words, when the older brother reached the finish, the younger brother had run \(97\) meters. They decided to race again, this time with the older brother starting \(3\) meters behind the starting line. Assuming that both boys ran the second race at the same speed as before, who do you think won?

### Meditating Monk

A monk needs to meditate for exactly forty-five minutes, but—living in an abbey—he doesn’t have a watch or a clock with which to time himself. All he has are two incense sticks, which he knows each take exactly one hour to burn. Unfortunately, being hand-made, the incense sticks aren’t identical to each other, and are imperfectly shaped so that he can’t rely on a stick burning at the same rate all the time.

Using these incense sticks and some matches, how can the monk arrange for exactly forty-five minutes of meditation?

### Untitled

Fifty natural numbers are written in such a way so that the sum of any four consecutive numbers is \(53\). The first number is \(3\), the 19th number is eight times the 13th number, and the 28th number is five times the 37th number. What is the 44th number?