# Engaging Questions

Last updated 10 April 2022, Mike Pierce

This is a collection of my favorite questions, typically classified as recreational mathematics or logic puzzles, that manage to engage students in a way the uninspiring exercises assigned in a usual math class will not. Email me if you know the original source of any of them. Also see my resources and links to similar collections.

1. 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: they can carry enough fuel to fly half-way around the planet, and they are capable of exchanging fuel instantaneously midair. What is the fewest number of planes that would have to be used to allow one plane fly completely around the planet and back to the small island along a great circle?

Paraphrased from Martin Gardner’s My Best Mathematical and Logic Puzzles.  |  PDF
2. Three friends Anita, Becca, and Charleston are challenged to a game by the Game Maestro. The Game Maestro places 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. He then places the three friends in a circle so that each of them can see the dots on their friends’ foreheads, but not on their own. The game proceeds like this: The Maestro 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 Maestro asks the next person. If someone responds “yes” and is right, the friends win! Whereas if someone responds “yes” and is wrong, all three friends will be banished to the shadow realm.

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?

3. 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?

Martin Gardner  |  PDF
4. Say that 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 pets of a particular kind.

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?

5. 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.”

6. 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.

7. 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.

8. 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?

9. 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?

Found on Andrej Cherkaev’s Math Puzzles page  |  PDF
10. 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?