$7\mathrm{.}1$ Shannon entropy $(6$ points$)$

You have $\backslash ($ N$=10\backslash )$ different samples of a material. They are supposed to have the same weight, but there's an odd one out. Your goal is to find which one is the odd one out, but all you have is a pan balance. You can load its sides with any number of samples, and it will tell you whether the left side weighs more, the right side weighs more, or they both weigh the same.

$1.$ Find the optimal loading strategy that maximizes the information gain of a single measurement. Plot the information gain $($in bits$)$ for different loading strategies.

$2.$ For the optimal strategy found above, how many measurements do you need to do on average to find the odd one out?

$3.$ What would be the information gain of a single $($optimal$)$ measurement in the limit of $\backslash ($ N $\backslash$rightarrow $\backslash$infty $\backslash )?$