Answers:

Solution to 1:

It helps to have a list of Fibonacci numbers in front of you when you start:

If you always use the biggest ones you can, they will also be non-consecutive (no two together)

a. 48 = 34 + 13 + 1

The biggest Fibonacci less than 48 is 34, so we will use that: 48=34+???

48-34=14. The biggest Fibonacci less than 14 is 13, so we will use that: 48=34+13+???

14-13=1 which is a Fibonacci number, so we use it to end our sum: 48 = 34 + 13 + 1

b. 41 = 34 + 5 + 2

The biggest Fibonacci less than 41 is 34, so we will use that: 41=34+???

41-34=7. The biggest Fibonacci less than 7 is 5, so we will use that: 41=34+5+???

7-5=2 which is a Fibonacci number, so we use it to end our sum: 41 = 34 + 5 + 2

c. 29 = 21 + 8

d. 33 = 1 + 3 + 8 + 13 + 21

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

a. 48 = 34 + 13 + 1, so take 1 counter (always take the smallest number in the Fibonacci sum).

b. 41 = 34 + 5 + 2, so take 2 counters

c. 29 = 21 + 8, so take 8 counters

d. 33 = 1 + 3 + 8 + 13 + 21, so take 1 counter

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?

53-1-2=50 counters

The Fibonacci sum for 50 is 34+13+3, so take 3 counters next (you are allowed to take up to 4)

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

31 - 2 - 1 = 28 counters now

The Fibonacci sum for 31 is 21+8+2, so take 2 counters next (you are allowed to take up to 2)

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

37=34+3. You are allowed to take up to 8. Take 3.

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

27=21+5+1. Take 1.

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

47=34+13. You are allowed to take up to 16. take 13.

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

7=5+2. You are allowed to take up to 8. You could win by taking 2. On the other hand, you'd win faster if you took all 7.

g. Your opponent just took 3 counters. There are 5 counters left. How many should you take? 5=5 (it's a Fibonacci). Happily, you can take up to 6 counters. Take all 5.

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)

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)

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 L_n+L_n+1=L_n+2

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

2,4,6,10,16

(so the sequence goes 0, 2, 2, 4, 6, 10, 16, ...)

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

6, 7, 13, 20, 33

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

L₉+L₁₀=L₁₁, so:
L₁₁=39+63=102

L₈+L₉=L₁₀
L₈+39=63, so
L₈=63-39=24

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

L₁₀+L₁₁=L₁₂, so:
L₁₁=118+191=309

L₉+L₁₀=L₁₁
L₉+118=191, so
L₉=191-118=73