Imagine a planet covered entirely with water with the exception of a single small island. On this small island is a sizeable fleet of airplanes and an endless supply of fuel. Now these airplanes are very peculiar: they can hold only enough fuel to fly half-way around the planet, and they are capable of swapping fuel nearly instantaneously midair. What is the fewest number of planes that would have to be used to get one plane to fly all the way around the planet and back to the island again along a great circle?

— Paraphrased from Martin Gardner’s My Best Mathematical and Logic Puzzles.

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?

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.

Without breaking the sticks, the monk lays the incense on the floor, lights it, and closes his eyes to meditate. Forty-five minutes later, he no longer smells the incense burning, and knows his meditation is complete. How did he manage this?

Suppose you are placing planes in three-dimensional space that all go through the origin $(0,0,0)$ with the goal of separating space into as many parts as possible. So when you place the first plane, you've separated space into two pieces. You can place a second plane and separate space into four pieces, and then you can place a third plane to separate space into eight parts (think of the coordinate planes as an example of this). If you were to place a fourth plane in space going through the origin, what is the largest number of parts that you can have divided space into? Then what if you place a fifth plane in space?

There's a map of the world that has every single town in the world marked on it. For each town on this map you draw a straight line connecting it to its nearest neighboring town (nearest in terms of distance along a straight line, not distance along roads). Show that after you've done this for every town on the map, each town can be connected to at most five others.

Fifty natural numbers are written in such a way so that 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?

Suppose that in the plane (or in $\mathbb{R}^2$ 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.

Suppose that $x$, $y$, $x-y$, and $x+y$ are all positive prime numbers. What is the sum of the four numbers?