3. Consider an equilateral triangle with side 700. Suppose that one move consists of changing the length of any of the sides of a triangle such that the result will still be a triangle. Find the minimum number of moves to change the given triangle to an equilateral triangle with side 2 .
Answer: 14
Solution: Work backwards by starting from $(2,2,2)$. The fastest way to do this is to lengthen the shortest side, making it as long as the sum of the other two sides. Denote $k^{-}$to be a real number arbitrarily close to $k$, but less than $k$.

First move: $\left(2,2,4^{-}\right)$, Second move: $\left(2,4^{-}, 6^{-}\right)$, Third move: $\left(4^{-}, 6^{-}, 10^{-}\right)$, and so on. Notice that it follows double of the Fibonacci sequence.
$$
2,2,4,6,10,16,26,42,68,110,178,288,466
$$

The side lengths after the 11 th move are $\left(178^{-}, 288^{-}, 466^{-}\right)$. After 3 moves, we can change all the sides to 700 as follows:
Step 12: $(288,466,700)$
Step 13: $(466,700,700)$
Step 14: $(700,700,700)$
Thus, the minimum number of moves is 14 .