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 the
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 friends
or mutually total 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,
or are 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 everyone in the village 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 other villager is highly logical and devout
(and they all know that they all know that each other villager
is highly logical and religiously devout, and …).
One day a blue-eyed mountaineer, tired and hungry,
stumbles across the village and asks for food.
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 you to tell me
whether 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 must lay 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.
Clock Hands
Between noon today and noon tomorrow, how many times
does the long hand on the clock pass the short hand?
“Pass” means that one hand follows, overtakes,
and goes ahead of the other.
Since both hands are at the same spot at noon,
the long hand does not pass the short hand
at twelve o'clock, the starting time.
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?
BdMO 2016
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?
Painting Cubes
Imagine that you have a can of red paint, a can of blue paint,
and a large supply of wooden cubes, all the same size.
You decide to paint the cubes by making each face either solid red or solid blue.
For example, you might paint one cube all red.
The next cube you may decide to give three red faces and three blue faces.
Perhaps the third cube can also be given three red and three blue faces,
but painted in such a way that it doesn’t match the second cube.
How many cubes can you paint in this manner that will be different from each other?
Challenge
Now imagine that instead of wooden cubes,
you are painting the faces of wooden objects
in the shape of one of the other platonic solids with more sides:
an octahedron (eight triangular sides),
an dodecahedron (twelve pentagonal sides),
or an icosahedron (twenty triangular sides).
How many possible ways are there to paint the solids
in each of these cases?
What Shape is the Planet?
Suppose you are on a large alien planet;
looking out around you, in every the direction
the surface of the planet curves gently, disappearing into the horizon.
But is this a “planet” as you know it?
How do you know this planet is spherical like your home planet Earth?
For all you know, this planet might be toroidal
— doughnut-shaped, with a big hole going through it.
Luckily, you have with you a paintbrush
and a can containing infinitely much red paint.
Using this, how can you tell whether or not the planet you’re on
is spherical or toroidal?
Challenge
Suppose you do discover the planet you’re on is toroidal and has a hole.
Can you be sure that there is only one hole?
In case there are more, can you use your paint and paintbrush
to determine how many holes this planet has?
Shielding Light with Spheres
Suppose that in three-dimensional space
there is a single point emitting light in every direction around it.
How many spheres (or any size) placed anywhere around this point in space
would be required to shield the reaches of space from this light?
