4. 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

5. 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)