2008 UIL State Test #17

[88bobcat]

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I understand there's a relationship between tiling (tesselation) and Fibonacci numbers. However, all the material I've seen talks about 1 x 1 or 1 x 2 tiles in rectangles.

#17 on the 2008 UIL State test has 3 x k sections.

MA08S_17.jpg (14.13 KiB) Viewed 1514 times

Are Fibonacci techniques used for solving this problem? Are there any good texts that I can reference to learn more about Fibonaccis in general? Where can I gain a more comprehensive understanding? I know the answer is (D) 41, but I had to manually count the squares.

Any guidance will be greatly appreciated.


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[AuSmith]

Well, I can't imagine any good general result covering a question like that. Exhaustion seems the way to go for me. As for Fibonacci numbers, Wikipedia has a wealth of information:



There are many relationships to do with the sequence. I believe only a few at a time come up on UIL tests, so I don't think UIL requires a deep understanding of Fibonacci, just the memorization of a few tricks. All I can say is that there are several facts hidden in Binet's formula,

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First, for the second parenthesized expression, [unparseable or potentially dangerous latex formula], meaning the Fibonacci sequence becomes practically exponential. Indeed, take the fifteenth term, [unparseable or potentially dangerous latex formula],

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That's about as much as anyone needs.

[88bobcat]

[stupidityismygam]

[88bobcat]

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