Last week, in Number Sense 038, Awkward Goat proved that a complete list of prime numbers could not be made, because there was no end to prime numbers: more could always be found and added to any supposed complete list of prime numbers. He didn't phrase it quite this way, but he proved there are an infinite number of primes. This week, the weather is unseasonably warm, daffodils are poking their buds above the ground, and spring is in the air.

“Ah, Spring!” sighed Billy Goat Gruff.

“Wonderful Spring,” agreed Awkward Goat, “when a goat's thoughts turn to...”

“Asserting dominance so I can get all the best nanny goats,” finished Billy Goat Gruff.

“That's fine for you, but sometimes I think Rabbits have the right idea,” opined Awkward Goat.

“What? Rabbits? Why on earth would you want goats to be like rabbits?” asked Billy Goat Gruff.

“Well, rabbits pair up,” said Awkward Goat.

“Think about it,” cautioned Billy Goat Gruff.

“All right,” replied Awkward Goat. “Suppose we start with one billy and one nanny. One pair.”

“And she has two kits,” said Billy Goat Gruff, “as usual.”

“Not right away,” said Awkward Goat. “They have to grow up first.”

“And then she has two kits,” said Billy Goat Gruff.

“Right,” said Awkward Goat, “and now we have two pairs of goats. A while later...”

“The kits are grown, and our first pair has had another pair of kits.” said Billy Goat Gruff, “and there are three pairs of goats.”

“And after that, the first pair has another pair of kits, the second pair of kits grow up, the grown up kits have their own baby goats,” said Awkward Goat.

“This is getting complicated,” complained Billy Goat Gruff.

“It is, isn't it,” replied Awkward Goat, “but I think I see a pattern. Each round, there are the same number of breeding pairs as there were total pairs the previous round.”

“Of course,” said Billy Goat Gruff. “Kits grow up. You can't expect them to be kits forever.”

“And the number of kit pairs in each round is the same as the pairs in the round before that. Each round is the sum of the two previous rounds. That means the next round will be the sum of this round plus the round before it.”

“We started with one pair,” said Billy Goat Gruff, “there was no previous round...”

“So the number of pairs in that non-existent round was zero,” said Awkward Goat.

“Zero plus one is one,” said Billy Goat Gruff. “Ok, that works out. The second round only had one pair as well.”

“Then one plus one is two, two plus one is three, three plus two is five...”

“And the next round should be five plus three is eight,” finished Billy Goat Gruff.

“Sure,” replied Awkward Goat. “All the goats from the previous round are grown up, but only the goats that were grown the previous round have kits. You're getting good at this, Billy Goat Gruff.”

“Thank you,” said Billy Goat Gruff. “The next round will be eight plus five, or thirteen pairs of goats. By the way, that's twenty six goats total. And one more round will give us thirteen plus eight or twenty one pairs, or forty two goats. Then twenty one plus thirteen is thirty four, thirty four plus twenty one is fifty five... That's over one hundred goats. One hundred hungry goats. What are all those goats going to eat?”

“Well, it's only one hundred and ten goats...” said Awkward Goat.

“But then fifty five plus thirty four is eighty nine! Then eighty nine plus fifty five is one hundred forty four! And that's starting with only one pair!”

“I guess breeding like rabbits is not such a good idea after all,” admitted Awkward Goat. “But wait! Rabbits breed like rabbits! Why aren't rabbits all over the place?”

“I've got two things to say about that,” replied Billy Goat Gruff, “In the first place, look around, rabbits are all over the place. And secondly: owls, hawks, coyotes, snakes, dogs, wolves, mountain lions... every time a rabbit turns around, he sees something that wants to eat him. We goats call them rabbits, those predators call them 'lunch'.  You're good at math, Awkward Goat, but I think you should stay away from social engineering.”

“Well, when you're right, you're right, Billy Goat Gruff,” said Awkward Goat, “and I thank you for helping me think this through.”

“You're quite welcome,” said Billy Goat Gruff.

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#### Comment Preferences

• ##### Tip Jar(10+ / 0-)

"The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

• ##### Maybe you should have had a guest goat(5+ / 0-)
Recommended by:
Orinoco, plf515, 2thanks, bnasley, Matt Z

Leonardo of Pisa today.

:)
Great stuff, thanks!

• ##### Fibogoatchi numbers!(5+ / 0-)
Recommended by:
Orinoco, 2thanks, palantir, bnasley, Matt Z

A long time ago, when I was in junior high (what they call middle school nowadays) I showed that you can start with ANY two numbers and get to the same ratio very quickly

210201  1   210202 210203 420406

and the ratios converge fast

It even works with negative numbers, and, although I didn't know it at the time, with complex numbers too.

I was all excited.  Then my math teacher told me that was great, but had been known for a couple centuries.

I write about Learning Disabilities and general stuff at Associated Content

• ##### Discoveries are discoveries.(5+ / 0-)
Recommended by:
plf515, 2thanks, palantir, bnasley, Matt Z

It really doesn't matter if someone did it before you, if you didn't know about it, it's also original to you. "It's been known for centuries" doesn't amount to a hill of beans if no one bothered to tell you.

The classic fibonacci series converges on phi to 3 decimal places in about ten steps. Is is quicker with larger numbers?

7+12=19            1.58333
12+19=31          1.63157
19+31=50          1.61290
31+50=81          1.62000
50+81=131        1.61728
81+131=212       1.61832
131+212=343     1.61792
212+343=555     1.61808
343+555=898     1.61801

Here is an example with two random numbers, converging on phi to 4 decimal places in 8 steps. Perhaps this one converged quicker because of the ratio of the first step?

"The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

[ Parent ]

• ##### It is certainly quicker with SOME larger numbers(4+ / 0-)
Recommended by:
Orinoco, 2thanks, palantir, Matt Z

I have a feeling it's related to how close the ratio of the first two is to phi. If you start with, say 13 and 21, it's like eliminating the first few steps of the "usual" series.

There's probably some way to get the rate of convergence from the initial ratio.

I write about Learning Disabilities and general stuff at Associated Content

[ Parent ]

• ##### talking about goats and rabbits multiplying,(4+ / 0-)
Recommended by:
palantir, Orinoco, bnasley, Matt Z

you are getting to a simple demonstration of carrying capacity in an environmantal niche.

With the rabbits, you have rates of predation limiting growth.

Simple graphs, simple curves to understand.

Thanks again, Orinoco.

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