Habits of Mind Problem #5
Fibonacci Follies
Due October 4, 2011
1. Suppose you are playing a first round of Fibonacci nim with a friend. You start with 15 counters.
You start by removing removing 2 counters; your friend then takes 1; you take 2; your friend
takes 1. What should your next move be? Can you make it without breaking the rules of the
game? Did you make a mistake at some point? If so, where?
Note that 15 = 13 + 2 = F6 + F2, so taking 2 is correct. Your friend takes 1 from 13, leaving 12.
Note that 12 = 8 + 3 + 1 = F5 + F3 + F1. Here is where you make a mistake, you should take 1,
instead you take 2, leaving 10. Now your friend takes 1, leaving 9. Since 9 = 8 + 1 = F5 + F1, so
you can recover by taking 1. You can still win, by considering that you are starting from 9,
always choosing the smallest Fibonacci number in the most economical representation as a sum.
2. Suppose you are playing a second round of Fibonacci nim with a friend. You start with 35
counters. You start by removing removing 1 counter; your friend then takes 2; you take 3; your
friend takes 6; you take 3; your friend takes 2. What should your next move be? Can you make it
without breaking the rules of the game? Did you make a mistake at some point? If so, where?
Note that 35 = 34 + 1 = F8 + F1, so taking 1 is correct. Your friend takes 2 from 34, leaving 32.
Note that 32 = 21 + 8 + 3 = F7 + F5 + F3 so you take 3, leaving 29. Now your friend takes 6,
leaving 23. Since 23 = 21 + 2 = F7 + F2, but you take 3 leaving 20! Here is where you make a
mistake, you should take 2, instead you take 3, leaving 20 . Now your friend takes 2, leaving 18.
Unfortunately, considering that you are starting from 18 = 13 + 5, you should take 5 to win, but
you can take at most 4! If your friend makes another mistake you may still be able to win, but not
guaranteed!
3. Suppose you are playing a third round of Fibonacci nim with a friend. You start with 21
counters. You start by removing removing 1 counter; your friend then takes 2. What should your
next move be? Can you make it without breaking the rules of the game? What went wrong?
Note that 21 = F7, so you are starting from a Fibonacci number. The strategy will not work!.
You take 1 from 21, leaving 20. Your friend takes 2, leaving 18. Note that 18 = 13 + 5 = F6 + F4.
You would like to take 5, but you can take at most 4. If your friend makes another mistake you
may still be able to win, but not guaranteed!
4. Suppose we create a new generalized sequence of numbers starting with 0 and 1 only this time,
instead of adding the two previous numebrs to get the next one, let's generate the next term by
adding 2 times the previous term to the term before it. In other words,
Gn+1 = 2*Gn + Gn-1.
Write a table of the first 15 generalized Fibonacci numbers.
Can every natural number be expressed as the sum of distinct, nonconsecutive generalized Fibonacci
numbers (as we did above)? Show why or if not possible, give three counter-examples, that is, three
natural numbers for which it is not possible. What if you were allowed to use consecutive generalized
Fibonacci numbers?
0
1
2
3
4
5
6
7
0
1
2
5
12
29
70
169
The gaps are too large, it is not possible to form 9,10, 11, 12, 13, 14, 15, or 16 as a sum of thse
generalized numbers, without repeats, even allowing for consecutive values.