  The guesswork refers to the distribution of the minimum number of trials
needed to guess a realization of a random variable accurately. In this study, a
non-trivial generalization of the guesswork called guessing cost (also referred
to as cost of guessing) is introduced, and an optimal strategy for finding the
$\rho$-th moment of guessing cost is provided for a random variable defined on
a finite set whereby each choice is associated with a positive finite cost
value (unit cost corresponds to the original guesswork). Moreover, we drive
asymptotically tight upper and lower bounds on the logarithm of guessing cost
moments. Similar to previous studies on the guesswork, established bounds on
the moments of guessing cost quantify the accumulated cost of guesses required
for correctly identifying the unknown choice and are expressed in terms of
R\'enyi's entropy. Moreover, new random variables are introduced to establish
connections between the guessing cost and the guesswork, leading to induced
strategies. Establishing this implicit connection helped us obtain improved
bounds for the non-asymptotic region. As a consequence, we establish the
guessing cost exponent in terms of R\'enyi entropy rate on the moments of the
guessing cost using the optimal strategy by considering a sequence of
independent random variables with different cost distributions. Finally, with
slight modifications to the original problem, these results are shown to be
applicable for bounding the overall repair bandwidth for distributed data
storage systems backed up by base stations and protected by bipartite graph
codes.
