On every June 1st, the Children's Day, there will be a game named "crashing balloon"
on TV. The rule is very simple. On the ground there are 100 labeled balloons,
with the numbers 1 to 100. After the referee shouts "Let's go!" the two players,
who each starts with a score of "1", race to crash the balloons by their feet and,
at the same time, multiply their scores by the numbers written on the balloons they
crash. After a minute, the little audiences are allowed to take the remaining
balloons away, and each contestant reports his\her score, the product of the numbers
on the balloons he\she's crashed. The unofficial winner is the player who
announced the highest score.
Inevitably, though, disputes arise, and so the official winner is not determined
until the disputes are resolved. The player who claims the lower score is entitled
to challenge his\her opponent's score. The player with the lower score is presumed
to have told the truth, because if he\she were to lie about his\her score, he\she
would surely come up with a bigger better lie. The challenge is upheld if the
player with the higher score has a score that cannot be achieved with balloons not
crashed by the challenging player. So, if the challenge is successful, the player
claiming the lower score wins.
So, for example, if one player claims 343 points and the other claims 49, then
clearly the first player is lying; the only way to score 343 is by crashing balloons labeled 7 and 49, and the only way to score 49 is by crashing a balloon laeled 49.
Since each of two scores requires crashing the balloon labeled 49, the one
claiming 343 points is presumed to be lying.
On the other hand, if one player claims 162 points and the other claims 81, it is
possible for both to be telling the truth (e.g. one crashes balloons 2, 3 and 27,
while the other crashes balloon 81), so the challenge would not be upheld.
By the way, if the challenger made a mistake on calculating his/her score, then the challenge would not be upheld. For example, if one player claims 10001 points and
the other claims 10003, then clearly none of them are telling the truth. In this
case, the challenge would not be upheld.
Unfortunately, anyone who is willing to referee a game of crashing balloon is
likely to get over-excited in the hot atmosphere that he\she could not reasonably be expected to
perform the intricate calculations that refereeing requires. Hence the need for
you, sober programmer, to provide a software solution.