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PUZZLE #8: CUSHIONING AN EGG

There are several versions of this on the web, but I thought the
following engineering scenario would provide some practical motivation.
You want to test a new cushioning material for automobile dashboards
by placing a sheet of the material at the base of a 100-foot
tower and dropping eggs onto it from various heights. Your goal
is to determine the maximum height X--in whole numbers of feet--from
which you can drop an egg and not have it smash to bits; and for
fun, you have set yourself the task of accomplishing this with two
eggs, in the smallest number of drops.
(You may assume that the eggs are structurally
identical, and that an egg which has survived a drop is unaffected and
may be reused.)

Of course you can get lucky, start with a drop from 100 feet and have
the egg survive,
but that is leaving too
much to chance. What you want to find is the smallest number N
for which you can
*guarantee* a determination of X in N drops. (And once you've
solved that problem, find the smallest number of drops with
which you can guarantee a determination of X with three eggs.)

(solution)