Last week, in Number Sense 039, Awkward Goat and Billy Goat Gruff explored Fibonacci series using Leonardo of Pisa's classic rabbit reproduction puzzle. They decided sauce for the goose wouldn't really work for the gander, besides, metaphors involving cooking farm animals left them feeling a bit queasy.

This week finds them wandering in a circle around a tree...

“What on earth are we doing, Awkward Goat?” asked Billy Goat Gruff.

“Why, we are walking around a tree,” replied Awkward Goat.

“Yes, yes, but why are we walking around a tree?”

“Let's suppose...” began Awkward Goat.

“Not this again!” said Billy Goat Gruff.

“...that we are walking along a number line,” continued Awkward Goat, undeterred by Billy Goat Gruff's bad attitude.

“And let us imagine a fence around the tree, with equally spaced fence posts.”

“Of course,” replied Billy Goat Gruff, “I see them now!”

“Well, then,” asked Awkward Goat, “how many are there?”

“Let's suppose,” said Billy Goat Gruff, “that there are twelve of them.”

“Oh very good,” replied Awkward Goat. “Let's number them.”

“Fine. This is number one, this is number two...”

“Wait! Have you forgotten this is a number line? What number do we start with on a number line?” asked Awkward Goat.

“Sorry, I didn't realize it was so important,” replied Billy Goat Gruff. “We start numbering with zero, so this is number Zero, this is number One, this is number Two, this is number Three...”

Billy Goat Gruff stopped abruptly.

“What's the matter?” asked Awkward Goat.

“That post already has a number,” replied Billy Goat Gruff. “What do we do now?”

“Since our circular number line has a fixed length,” replied Awkward Goat, “we are finished numbering the posts. It's not like an ordinary number line which goes on forever in both directions. We can call it a number circle.”

“But what good is it?” asked Billy Goat Gruff. “I can use an ordinary number line for addition and subtraction and multiplication. This circular number line doesn't even have negative numbers!”

“That's true,” said Awkward Goat, “and it seems we don't need them. If we turn around and walk the other way, which can be considered going in the negative direction, we will always get to some numbered post, which is the answer to any subtraction problem.”

“So if we go forward, say, to two, then subtract a larger number, say four, by going the other way, we would ordinarily get a negative number. Two minus four is negative two. But on this circular number line we get ten! How on earth does that make sense?” asked Billy Goat Gruff.

“What time is it?” asked Awkward Goat.

“Don't dodge the issue,” said Billy Goat Gruff. “Answer my question.”

“Humor me,” Awkward Goat requested.

“All right. It's two o'clock,” replied Billy Goat Gruff.

“And we've been walking around this tree for how long?”

“Four bloody hours!” answered Billy Goat Gruff, “more or less.”

“And that means we got here...” began Awkward Goat.

“At ten o'clock this morning!” finished Billy Goat Gruff.

“So two o'clock minus four hours is...”

Have fun in the comments.

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