I seem to have some explanatory pages I came up with long ago. Hopefully they will be helpful and not confusing. Here is one on Fibonacci sums, and one on Fibonacci NIM

Practice problems :

1. Find the way to write each of these numbers as the sum of non-consecutive Fibonacci numbers

a. 48

b. 41

c. 29

d. 33

2. For a game of Fibonacci NIM, starting with the number of counters below, what should the winning first move be?

a. 48

b. 41

c. 29

d. 33

3. For a game of Fibonacci NIM, what should your next move be to win?

a. Start with 53 counters, you take 1, your opponent takes 2. How many should you take next?

b. Start with 31, you take 2, your opponent takes 1. How many should you take next?

c. Your opponent just took 4 counters; there are 37 counters left. How many should you take?

d. Your opponent just took 2 counters, there are 27 counters left. How many should you take?

e. Your opponent just took 8 counters. There are 47 counters left. How many should you take?

f. Your opponent just took 4 counters. There are 7 counters left. How many should you take?

g. Your opponent just took 3 counters. There are 5 counters left. How many should you take?

4. You are playing Fibonacci NIM with Emil Ennui. If he can't make a move that would let him win, he always takes the maximum number of counters he is allowed to take. Show me what would happen if you play a game starting with 53 counters would go like (your turn first)

# counters at beginning of round Whose turn How many counters are taken
53 me  
  Emil  
  me  
  Emil  
  me  
  Emil  
...    

 

5. You are playing Fibonacci NIM with Sam Sleepy. If he can't win the game by taking all of the counters, he always takes 2. Show me what would happen if you play a game starting with 28 counters (you go first)

# counters at beginning of round Whose turn How many counters are taken
28 me  
  Sam  
  me  
  Sam  
  me  
  Sam  
...    

6. Lucas numbers have the same rule for getting the next number as Fibonacci numbers, but they start with different numbers. The Lucas number rule (like the Fibonacci rule) is Ln+Ln+1=Ln+2

A. If you start a Lucas sequence with 0, 2,... what are the next 5 numbers in the sequence?

B. If you start a Lucas sequence with 5,1,... what are the next 5 numbers in the sequence?

C. In a particular Lucas sequence, L9 = 39 and L10 = 63. Use these values to find L8 and L11

D. In a particular Lucas sequence, L10 = 118 and L11 = 191 . Use these values to find L9 and L12

Answers

Play Fibonacci NIM online here