Remember Bob the absent$-$minded student? Bob has a set of $6$ passwords

that he uses for all his login needs. He o$$en forgets which password matches with

which website so his strategy is to try all of them. Specifically, from his set of pass$-$

words, he picks one uniformly at random and enters it$.$ If he succeeds $$ all is well; if

he fails $$ he removes the password from the set and repeats the process. He keeps

entering an unused password un$$l he succeeds.

To help with the analysis, let$$s represent Bob$$s passwords as $\backslash$alpha $1,\backslash$alpha $2,\backslash$alpha $3,\backslash$alpha $4,\backslash$alpha $5,\backslash$alpha $6.$

Let$$s also assume that the password that will let Bob enter a website is $\backslash$alpha $3$ but of

course Bob has forgo$$en this fact.

It turns out that the website has a policy that if a user fails to enter the correct

password for k consecu$$ve $$mes$,$ it will delay logging in the user by $2$k seconds. For

example, if Bob gets the password right the first $$me$,$ Bob experiences a delay of

$20=1$ second before he is logged on to the website. If Bob gets the password right

the second $$me$,$ he experiences a delay of $21=2$ seconds before he is logged on to

the website, etc.