Make sure you include referencing for answers where it would obviously be needed.

$1.$ You have two puzzles with parameters as follows:

Puzzle A: One sub$-$puzzles. $\backslash ($ k$=6\backslash ).$

Puzzle B: Six sub$-$puzzles. $\backslash ($ k$=4\backslash ).$

You should provide, for both cases other than part $($b$),$ the following:

$($a$)$ The distribution of the number of cases that require each number of hashes.

$1\mathrm{.}0$ Mark

$($b$)$ Explain the method you used to obtain your distributions. Don't go into too many details or show working, it's more "I wrote a C$++$ program to $...$ and then using $...$ I $...".$

$0\mathrm{.}5$ Mark

$($c$)$ A graph of the distribution of the data above.

$0\mathrm{.}5$ Mark

$($d$)$ The average number of hashes needed.

$0\mathrm{.}5$ Mark

$($e$)$ The standard deviation for the distribution of the number of hashes needed.

$0\mathrm{.}5$ Mark

You should assume that if there are $\backslash ($ N $\backslash )$ possible solutions you check the $\backslash ($ N$^\{\backslash$text $\{$th $\}\}\backslash )$ by hashing even if all others have failed and there has to be a solution.