The Story
Once upon a time there was a little old lady who lived in a little old house in a little old wood, not too far from a field that stretched across a long hillside.
It was winter. Cold. The snow was deep. Her mind and body were going to seize up if she didn’t get some fresh air and exercise.
She put on showshoes and set out for the hillside. She walked all the way around the field, making a circle. She wasn’t tired. So instead of finishing the circle and going home, she aimed herself inward a bit and traced a smaller circle inside the first one, and then a smaller circle inside of that. And another. And so on, spiraling toward the middle of the field. She told herself circle jokes. Two Pie Arrrr! Pie Are Squared.
The circles got smaller. And smaller. Before long, she found herself in the middle.
She turned, let her steps trace an S, and started winding her way out.
It was a long way.
The circles got bigger and bigger. She thought about how much farther it is to walk all the way around when she could just take a short cut and go straight across. She was tired. She was tempted. But she didn’t want to mess up that spiral path.
She did the same thing the next day. And the next. She walked that snowy path a lot. Sometimes in the mornings. Sometimes in the moonlight. It reminded her of neighborhood games of Fox-and-Geese.
She told herself jokes about hamster wheels and crop circles. She thought about Albert Einstein’s theory of gravitation which she’d learned about in high school. They were supposed to think about spacetime and gravity by pretending you were watching the lights of dozens of people walking a spiral road up a mountainside in the dark, all of them carrying candles…
She decided to invite the townspeople to try this out. She would leave out the Einstein stuff and tell them it was a party to celebrate the new moon. And to bring a candle.
The Problem
The townspeople of her acquaintance were little old men and little old ladies like herself. She could see them already, forsaking their walkers for ski poles and launching themselves into the snow. Stamina would be overestimated. When they got tired, they would wreck the spiral by walking straight off the field towards food, grog and rest.
There had to be a warning flag. A flag placed somewhere on the trail that would signal, “Hey! Think about how far you’ve already come! Can you really make it all the way to the middle and back out? Do you think you should turn around now?”
The perplexing question
At what point on the path should she place the flag?
Figuring it out
She pondered the skills and tools she might need to figure this out.
- Measuring?
- Counting?
- Sunlight and shadows?
- A way to remember stuff?
- A stopwatch?
- An assistant?
- A team?
She drew these two pictures. Are they any help?
