Skip to content

Stories that never end

Infinity and strange math · level 1, lesson 5 of 6

A question to start

Here is a very old story. Read it out loud.

It was a dark and stormy night. The sailors sat round the fire, and the captain said: "Tell us a story, Sailor Sam." And Sailor Sam began:

"It was a dark and stormy night. The sailors sat round the fire, and the captain said: 'Tell us a story, Sailor Sam.' And Sailor Sam began: ..."

What happens next? How does this story end? If you kept reading it out loud, when would you get to the last word?

The idea

A story with an ending

Most stories have a beginning, a middle and an end. The three bears come home, they find Goldilocks, she runs away, the end. When you reach "the end", you close the book. There is a last page.

A story that starts itself again

The sailor story is different. Look at what happens in it. Sailor Sam is asked to tell a story, and the story he tells is the very same story. So when you reach the end of it, you are sent right back to the start: "It was a dark and stormy night ..."

We can draw this with boxes and arrows. Each box is a part of the story. Each arrow says "then comes".

It was a darkand stormy night The captain asksfor a story Sam begins thesame story ...

Follow the arrows with your finger. Is there a box with no arrow going out? No! Every box has an arrow to the next one, and the last one points back to the first. There is no box that says "the end".

That is why the story never finishes. There is no last page. It is a loop, like the circle in lesson 3. You can go round and round it forever.

How long would it take?

Say one telling of the sailor story takes about one minute to read out loud.

  • One telling: 1 minute.
  • Two tellings: 2 minutes.
  • Ten tellings: 10 minutes.
  • Sixty tellings: 60 minutes, which is one whole hour.

However long you keep going, there is always one more telling to do. The number of tellings is the counting numbers again: 1, 2, 3, ... and they never end. So the story would take forever to finish, which means it never finishes at all.

Of course a real person gets tired, or hungry, or bored, and stops. But that is the person stopping, not the story ending. The story itself never says "the end".

A story inside a story

There is a second strange thing about the sailor story. Sailor Sam is inside the story. And inside his story there is another Sailor Sam, telling the story again. And inside that story, another Sailor Sam ...

This is just like the mirrors in lesson 4: a picture inside a picture inside a picture. Here it is a story inside a story inside a story. Each Sam is one step deeper. There is no deepest Sam, because every Sam starts the story again.

A song that loops: the hole in the bucket

There is an old song about Henry and Liza. Henry has a bucket with a hole in it. Liza tells him to fix it. Here is what happens, in short:

  1. To fix the hole, Henry needs some straw.
  2. To cut the straw, he needs a sharp axe.
  3. To sharpen the axe, he needs to wet the sharpening stone.
  4. To wet the stone, he needs water.
  5. To fetch the water, he needs ... the bucket!
  6. But there is a hole in the bucket. So back to step 1.

Each step needs the next step, and the last step needs the first. Henry is stuck in a loop. The song can go round and round, and Henry never gets his bucket fixed.

Ways out of a loop

A loop never ends by itself. To get out, someone has to add a stopping rule. For example:

  • "Tell the sailor story three times, then stop."
  • "Sing the song until the bell rings."
  • "Run round the track 4 laps."

A stopping rule is like a door in the loop. Without the door, you go round forever.

There is a famous joke about this. Shampoo bottles sometimes say:

Wash, rinse, repeat.

"Repeat" means "do it again". So you wash, rinse, and do it again: wash, rinse, and ... do it again! If you followed these words exactly, you would be in the shower forever. The bottle forgot the stopping rule. It should say "repeat once".

Growing stories

Some stories do not loop. They grow: each part adds one more piece, then says all the old pieces again. The House that Jack Built is like this:

This is the house that Jack built.

This is the malt that lay in the house that Jack built.

This is the rat that ate the malt that lay in the house that Jack built.

Each verse is one line longer than the one before. The real rhyme stops after a few more verses. But you could always invent one more ("This is the cat that chased the rat ..."), so a growing story could go on forever, getting longer and longer. That is the "one more" trick from lesson 1.

Worked examples

1. Find the loop. "Pete and Repeat were in a boat. Pete fell out. Who was left?" The answer is "Repeat". And "repeat" means "say it again", so you say: "Pete and Repeat were in a boat. Pete fell out. Who was left?" The answer again is "Repeat" ... The end of the joke sends you back to the start. It is a loop.

2. Count the tellings. A loop story takes 2 minutes to tell once. How many times can you tell it in 10 minutes? Count in twos: 2, 4, 6, 8, 10. That is 5 tellings in 10 minutes. And after that, there is still one more telling waiting.

3. Add a stopping rule. Change the sailor story so it ends. One way: "And Sailor Sam began: 'It was a dark and stormy night.' Then he yawned, and everyone went to sleep. The end." Now the last box has no arrow going back. The loop is broken. The story has a last page.

4. Count the lines. In The House that Jack Built, verse 1 has 1 line, verse 2 has 2 lines, verse 3 has 3 lines. How many lines do verses 1, 2 and 3 have together? \(1 + 2 + 3 = 6\) lines. A growing story gets long fast.

Try it

  1. Does the story of the three bears have an end? Does the sailor story have an end?
  2. Draw boxes and arrows for the hole in the bucket song. Where is the arrow that makes it a loop?
  3. A loop story takes 3 minutes to tell once. How many minutes do 4 tellings take?
  4. Add a stopping rule to "Wash, rinse, repeat" so that you only wash twice.
  5. In a growing story, verse 1 has 1 line and each new verse has one more line. How many lines does verse 6 have?
  6. Ella says: "I told the sailor story 100 times, so now it is finished." Is she right?
  7. Make up your own loop story with three parts. Check that the last part leads back to the first.
Answers
  1. The three bears story has an end: Goldilocks runs away, the end. The sailor story has no end: it always starts again.
  2. Six boxes: hole in the bucket, then straw, then axe, then stone, then water, then bucket. The arrow that makes it a loop goes from "fetch the water with the bucket" back to "there is a hole in the bucket".
  3. Count in threes: 3, 6, 9, 12. Four tellings take 12 minutes.
  4. For example: "Wash, rinse, repeat once." Or "Wash, rinse, wash, rinse, stop." Then you wash twice in all.
  5. Verse 1 has 1 line, verse 2 has 2, and so on, so verse 6 has 6 lines.
  6. No. After 100 tellings, there is still a 101st telling waiting. The story never says "the end". Ella stopped, but the story did not finish.
  7. Many answers are right. For example: "A bird is hungry. It looks for a worm. It finds a worm, eats it, and flies off. Later, the bird is hungry ..." The last part leads back to the first, so it is a loop.

Common mistake

Thinking a loop story ends when the teller stops. The teller stops because they are tired. The story itself never reaches "the end". In math, "never ends" is about the rule, not about how long someone keeps going.

Mixing up a loop and a growing story. A loop comes back to the same place again and again. A growing story gets longer each time. Both can go on forever, but in different ways.

Where it leads