A few related problems…

Here is a collection of interesting math problems. Despite appearances, they all have something in common. Can you figure out what it is?

  • A single pair of baby rabbits is placed on an island. They take one month to grow up; after that, they have one new pair of baby rabbits every month. Each new pair of baby rabbits takes a month to grow up, then start having baby rabbits of their own, and so on. How many pairs of rabbits are on the island after n months?
  • Susie can climb either one or two steps at a time. How many different ways can Susie climb a staircase with n stairs? (For example, she could climb a staircase of 3 stairs in 3 different ways: 1-2, 2-1, or 1-1-1.)
  • How many different ways are there to tile a 2xn rectangle with 2×1 dominoes?
  • If male bees (drones) are produced asexually from a female (queen), but female bees have two parents, one male and one female, how many ancestors does a male bee have in the nth generation before it?

Well, ok, I sort of gave it away with the tags…believe it or not, the solution to each problem is the Fibonacci numbers! For each problem, can you figure out why? Can you come up with your own problem which has the Fibonacci numbers as the solution?

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3 Responses to A few related problems…

  1. Jonathan says:

    I artificially connect the rabbits and the stairs with this classic: Ghost the Bunny. I use it every year, with almost every class. I have occasionally used the dominoes, later, as much for the oohs and aahs when they notice the correspondence as for anything else.

  2. Brent says:

    Ah, I like the formulation with a bunny instead of a person. It seems somehow more “realistic” (whatever that means).

  3. Jonathan says:

    “Realistic” Hmm. If it means mockingly self-referential, then yes. But if “realistic” is meant in any way to be related to “real world,” I don’t think so.

    Maybe this year I’ll modify it (to make this clear): “Laura’s stuffed bunny, Ghost, hops up a flight of ten stairs…”

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