1.If you always use the biggest Fibonacci's you can, you will get the best sum.

a. 48 = 34 + 13 + 1

(How to get it:
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

How to get it:
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 left at this point

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 left now

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

4. You are playing a game of Fibonacci NIM against Sam Sleepy. You start with 28 counters. Sam always takes 2 tokens every turn*, unless he could win by taking more that turn. If you play using the best possible strategy, give a step by step accounting of what happens in the game.

If you write down a way to win that does not use the Fibonacci sums, you will not get credit, since it does not count as the best possible strategy.