MathDB

Two dueling wizards are at an altitude of

100

above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by

a

for me and by

b

for my rival" where

a

and

b

are real numbers such that

0 < a < b

. Different spells have different values for

a

and

b

. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells such that the second wizard has a guaranteed win, if the number of spells is

(a)

finite;

(b)

infinite?

Problem Statement