\def\gauss{e^{t^2 \over 2\sigma^2}} Entropy is the (logarithm of the) number of states of a system for which some \emph{macroscopic} measure is left invariant. The logarithm is there just so we can add entropies, instead of multiplying the number of possible states. % weighted sum? For example take 100 coins. If we describe the \emph{macrostate} by the number of heads, states with high entropy are those around 50, for which there are a huge number of possible \emph{microstates} leading to the same measure. States around 0 and 100 are low entropy, since there are only few possible such arrangements. Note that the definition of entropy depends on which measure(s) we are using to describe a macrostate. Quick, some airplane notes. Maximum entropy distribution for the interval $[0,1]$. Suppose the solution is $p(x) = 1$. The entropy is then given as $H_p = \int_0^1 1 \log 1 = 0$. We take another distribution $q(x) = 1 + \epsilon q'(x)$ with $\int_0^1 q'(x) = 0$. Using the linear approximation $\log(x) \approx 1 + x$, the linear approximation of the entropy is $H_q \approx - \int_0^1 (1 + \epsilon q'(x)) \epsilon q'(x) = - \int_0^1 \epsilon^2 q'(x)^2$, which is always less than $0$, so $p(x)$ has maximal entropy.