Obsolete Machine

It's important to note that 2016 is divisible by 16, so the last steps have to be ×2, ×2, ×2, ×2, +2. $2016 \div 16 = 126$ . This is perfect because we can get to 128 by powers of 2, then subtract 2 and then do the previously described last steps for a total of $\boxed{13}$ .

Here they are:

$\begin{aligned} 0 & \stackrel{+2}{=} 2 \ & 1 \\ & \stackrel{\times 2}{=} 4 \ & 2 \\ & \stackrel{\times 2}{=} 8 \ & 3 \\ & \stackrel{\times 2}{=} 16 \ & 4 \\ & \stackrel{\times 2}{=} 32 \ & 5 \\ & \stackrel{\times 2}{=} 64 \ & 6 \\ & \stackrel{\times 2}{=} 128 \ & 7 \\ & \stackrel{-2}{=} 126 \ & 8 \\ & \stackrel{\times 2}{=} 252 \ & 9 \\ & \stackrel{\times 2}{=} 504 \ & 10 \\ & \stackrel{\times 2}{=} 1008 \ & 11 \\ & \stackrel{\times 2}{=} 2016 \ & 12 \\ & \stackrel{+2}{=} \boxed{2018} \ & \boxed{13} \end{aligned}$

Since $2018=2048-32+2$ , we can make it by doing
$\begin{aligned} 0&\stackrel{+2}{\rightarrow}2 \qquad &&1\\ &\stackrel{\times2}{\rightarrow}4 &&2\\ &\stackrel{\times2}{\rightarrow}8 &&3\\ &\stackrel{\times2}{\rightarrow}16 &&4\\ &\stackrel{\times2}{\rightarrow}32 &&5\\ &\stackrel{\times2}{\rightarrow}64 &&6\\ &\stackrel{\times2}{\rightarrow}128 &&7\\ &\stackrel{-2}{\rightarrow}128-2 &&8\\ &\stackrel{\times2}{\rightarrow}256-4 &&9\\ &\stackrel{\times2}{\rightarrow}512-8 &&10\\ &\stackrel{\times2}{\rightarrow}1024-16 &&11\\ &\stackrel{\times2}{\rightarrow}2048-32 &&12\\ &\stackrel{+2}{\rightarrow}2048-32+2 &&13.\\ \end{aligned}$

Enter in a calculator to see: 0 +2 *2 *2 *2 *2 *2 *2 -2 *2 *2 *2 *2 +2, except each +, * or - is separate.

Count the $2$ s in $\text{IntegerDigits}$ 's first argument: $\text{IntegerDigits}[((0 + 2) 2\ 2\ 2\ 2\ 2\ 2 - 2) 2\ 2\ 2\ 2 + 2,2]={11111100010}_2={2018}_{10}$ . The count is 13.