image by Klearchos Kapoutsis. CC BY 2.0

People often think math is a rigid subject with a bunch of rules and tricks. But mathematicians are actually flexible thinkers, often imagining unseen forces and spirits to help solve problems.

In this Halloween edition on the Monday Puzzle, we present 4 math puzzles that can be solved by imagining ghostly forces.

(Regular readers can skip to puzzle 4 as I have posted puzzles 1-3 in past columns).

Puzzle 1: A father left 17 goats to his three sons. His will promised 1/2 of the goats to the oldest son, 1/3 to the middle son, and 1/9 to the youngest son. But the sons could not divide 17 evenly. How did they split up the goats?

Puzzle 2: You have several identical bricks and a ruler. You want to measure the diagonal of a single brick without using any formulas. How can you do it?

Puzzle 3: At 6 am a monk started climbing a mountain and reached the top at 8 pm, where he slept for the night. The next morning at 6 am he descended the mountain on the same path and reached the bottom at 8 pm. A mathematician made an interesting observation: by the facts, there must be some spot on the mountain that the monk occupied at exactly the same time of day for both trips. Why is this?

Puzzle 4: There are 100 ants on a 1 foot stick. The ants move left or right randomly at 1 foot per minute. Whenever two ants meet, each turns around and moves in the opposite direction. An ant that reaches the end of the stick falls off. Certainly if you wait long enough, all of the ants will fall off the stick. Is there a minimal time you can be sure the ants are all off the stick?

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Answer to Mathematicians Do Believe in Ghosts

Puzzle 1: Dividing 17 goats

A wise man told the sons to imagine a ghost goat so there would be 18 goats in total. By the will, the oldest son would get 1/2 of the total 18 which is 9 goats. Then the middle son would get 1/3 of the total 18 to get 6 goats, and the youngest would get 1/9 of the total 18 to get 2 goats.

This division means the three sons have 17 goats between them, as 17 = 9 + 6 + 2. After serving its purpose, the ghostly goat could leave the physical world and disappear.

Puzzle 2: Three Bricks

You only need 3 bricks for this to work. Place one brick on top of another, and place the third brick next to them. This will create a space for an invisible fourth brick.

You can directly measure the diagonal of this “ghost” brick.

Puzzle 3: Monk’s Journey

The monk’s journey and be solved by plotting the altitude up and down on a time scale, and then invoking a fixed point theorem that guarantees the two paths cross. But there is a trick by imagining a ghost.

Imagine the two journeys take place on the same day. In other words, while the monk is descending at 6am, imagine a ghost traces out the path of his journey up the mountain at 6am from the day before. Obviously the monk going down has to cross the ghost of the monk coming up! This spot is where the monk occupies the same location at the exact same time on both days.

Puzzle 4: Ants on a Stick

When two ants collide, they turn around and walk in the opposite directions. Mathematically, this is exactly the same as if the two ants simply passed through each other like ghosts.

The longest distance a ghost ant has to travel is starting at one end of the stick and moving to the other. The ant can travel 1 foot in 1 minute, and therefore in 1 minute we can be sure all the ants have fallen off the stick.