#### A puzzle

Share this puzzle with family and friends:

- Gork and Zork are 8 steps away from their spaceship.
- They walk halfway to the spaceship (4 steps).
- Then they walk half the remaining distance (2 steps).
- Then they walk half the remaining distance (1 step).
- Then they walk half the remaining distance (1/2 step).
- Then they walk half the remaining distance (1/4 step).
- If they keep repeating this process.
- Will they ever get to the spaceship?

Did this puzzle surprise you?

How could Gork and Zork ever reach their spaceship?

#### Zeno

The ancient Greek philosopher Zeno maintained that it was impossible for someone to walk to the end of a path because they need to pass through the following distances:

- Half of the path distance,
- half of the remaining path distance,
- half of the remaining path distance,
- and so on

This requires to complete an infinite number of tasks, which, Zeno states, is impossible. This is a paradox, that is, valid reasoning with a logically unacceptable conclusion.

#### Infinity in your hand

Start with five 16×16 square grid (get the grid paper pdf file).

Shade the grids to represent the fractions 1/2, 1/4, 1/8, 1/16, 1/32

You will end up with something like this:

Imagine doing this forever (you need a good imagination!)

Now imagine cutting out all of the infinite shaded pieces and joining them together to form a single shape. How big would the new shape be?

- Would it fit in your school?
- Would it fit in your classroom?
- Can you hold it in your hands?

#### Geometric series

The sum of fractions we’ve been playing with is a *series*. A series is the sum of numbers in a pattern. Our pattern is that we get the next term by multiplying the previous term by 1/2.

Some series are *arithmetic*, like 2 + 4 + 6 + 8, where we add (or subtract) the same number. Some series, like ours, are *geometric*, where we multiply (or divide) by the same number. Also, some series are finite and some infinite. Our series is infinite.

Can an infinite series have a finite sum? Is there a sum for our series?

Watch Western University mathematician Graham Denham talk about this puzzle in the following videos:

**What did you learn?**

What did you learn about infinity?

- What surprised you?
- What math insights did you experience?

Share your learning with others, so they can experience math surprise and insight as well.