Jimmy Camps Just Outside the Fence… Or Does He?

Why seeing everything doesn’t tell you where you are

The Outdoor Education department of the school had recently taken Jimmy and some other students from my class on a three-day camp. I thought my lesson for the day might click with them.

“Jimmy,” I said, turning to the board, “imagine you’re out camping.”

“Already better than school,” he replied.

“You’re walking through the bush, and you come across this fenced area.”

I used my laptop to project a diagram onto the whiteboard.

It was one long, smooth continuous curve, more like a weather map than a fence.

“There’s a KEEP OUT! sign somewhere along the boundary,” I added.

“That doesn’t look like a fence, sir,” Jimmy commented.

“Good,” I said. “Because it isn’t about the fence. It’s about the boundary.”

Jimmy pointed to the red dot.

“That’s me, I suppose?”

“Yes. You’re standing right there. The question is — are you inside or outside?”

Jimmy didn’t answer straight away. He was studying the diagram.

“Well…” he said slowly, “there’s boundary on this side…”

He traced one direction with his finger.

“…and there’s boundary on the other side. So, I’m both.”

A few students laughed.

“Both?” I spoke.

“Yes,” Jimmy replied. “I’m inside one part… but outside another part. Depends which way you look.”

“That’s a very natural answer,” I said. “And also completely wrong.”

I drew a straight horizontal line from the red dot across the diagram to the right.

“Count how many times this line crosses the boundary.”

Jimmy traced it carefully.

“One… two… three… four… five… six.”

He stopped.

“Six crossings.”

“Odd or even?” I asked.

“Even. Why does that matter?”

“Because every time your line crosses the boundary, you change sides.”

I pointed far to the right.

“Imagine starting way out here, where you are definitely outside. Now walk along the line toward the red dot.”

I tapped the first crossing.

“First crossing: outside becomes inside.”

Then the second.

“Second crossing: inside becomes outside.”

“Each crossing flips me,” I emphasised.

Everyone understood.

Jimmy started counting again.

“One crossing: inside. Two crossings: outside. Three crossings: inside. Four crossings: outside…”

“If the number of crossings is odd, I end up inside,” he concluded.

“And if it’s even, you end up outside,” I said.

“And I got six, so I’m outside,” Jimmy observed.

Now he had a new thought.

“You drew the line to the right. Why not to the left?”

“Try it,” I said.

Jimmy traced a line to the left.

“One… two. Still even,” he stated.

But he wasn’t done.

“What about up?” he said. “Or down? Or diagonal?”

“Well, pick any straight direction you like,” I challenged him.

Jimmy imagined a line going upwards this time, then diagonally.

“Still even,” he said. “So it doesn’t matter?”

“Not if you do it properly,” I said. “Every time your line crosses the boundary, you flip between inside and outside. No matter which direction you choose, the total number of flips — odd or even — ends up the same.”

Jimmy thought about that.

“So I can’t cheat it by picking a different direction?”

“No.”

“Even if I go diagonal?”

“No. The answer isn’t in the direction, it’s in the boundary,” I stated.

Jimmy wasn’t finished.

“Sir… what if the fence overlaps itself?”

“What do you mean?” I asked.

“Like if the lines cross over each other,” he said, tracing two imaginary curves. “Or double back. Or sit on top of each other.”

“Ah,” I said. “Now you’re asking a mathematician’s question.”

“The rule we’ve been using — counting crossings — works perfectly when the boundary is a simple closed curve.”

“A what?” Jimmy and several others exclaimed.

“A boundary that doesn’t cross itself, doesn’t overlap, and cleanly separates inside from outside,” I explained.

I drew a quick example where a curve looped over itself.

“Now,” I said, “when your line crosses the boundary, you still flip sides — but the idea of ‘inside’ can become ambiguous.”

Jimmy frowned.

“So I could be half inside?”

“Not quite,” I said. “But different rules may be needed to decide what counts as inside.”

“So the simple odd–even rule might not work?” Jimmy asked.

“It still works in many cases,” I said, “but you have to be more careful about what you mean by ‘inside’.”

“Even fences have rules,” Jimmy noted. “The maths works as long as the fence behaves itself.”

“Anyway, I’d hate to be caught on the odd side of the law,” he added philosophically.