Consider the following game, in which a referee picks a random integer between 1 and 100.
One after the other, each of three players tries to guess the number the referee picked. Each
player announces his or her guess before the next player guesses. Each guess has to be different
from the previous guesses. The winner is the player who comes closest to the referee’s
number without exceeding it. (It is possible for none of the players to win.)
Suppose that Player 1 guesses 24, and that Player 3 will guess a number that gives her/him the
best chance of winning. What number should Player 2 guess to maximize his/her chances of
winning? Select:
A) 1
B) 25
C) 62
D) 63
E) 64
First of all, observe that it always gives the third player the best chances to win if   [s]he guesses the lowest number in some range which the other players left open. Thus, if you   choose 63, this narrows the third player’s choices down to 1, 25, and 64. Guessing 25 gives   a range of 38 numbers (25 through 62 inclusive) which win for the third player if the referee   picked one of them, while a choice of 64 would only give 37 favorable numbers, and a choice   of 1 only gives 23 favorable numbers.   Note that 63 gives you 38 favorable numbers also (63 through 100 inclusive) so this way you   and the third player both have a 38% chance of winning, while the first player only has a 24%   chance.   The other four given answers do not give you such a good result. Answer (e), 64, still makes   25 the best choice for the third player, but cuts your chances down to 37% while raising the   third player’s to 39%. Answer (a), 1, is even worse than (e) because it only gives 23 chances   for you. Answer (c), 62, would give the third player 63 as the best guess, (38% versus 37%   if 25 is guessed instead) drastically reducing your winning chances to 1%. You would get the   same baleful result if you guessed 25 [answer (b)], because then the obvious best guess for   the third player is 26.   correct answer 63
<result>D</result>
Consider the following game, in which a referee picks a random integer between 1 and 100.
One after the other, each of three players tries to guess the number the referee picked. Each
player announces his or her guess before the next player guesses. Each guess has to be different
from the previous guesses. The winner is the player who comes closest to the referee’s
number without exceeding it. (It is possible for none of the players to win.)
Suppose that Player 1 guesses 24, and that Player 3 will guess a number that gives her/him the
best chance of winning. What number should Player 2 guess to maximize his/her chances of
winning?