Download Problem: Alice and Bob play the following number game. Alice

Problem:
Alice and Bob play the following number game. Alice writes down a list of
positive integers x1 , x2 , ..., xn but does not reveal them to Bob, who will try to
determine the numbers by asking Alice questions. Bob chooses a list of positive
integers a1 , a2 , ..., an and asks Alice to tell him the value of x1 a1 + x2 a+, ... +
xn an . Then Bob chooses another list of positive integers b1 , b2 , ..., bn and asks
Alice for x1 b1 + x2 b2 +, ... + xn bn . Play continues until Bob is able to determine
Alice’s numbers. How many rounds will Bob need in order to determine Alice’s
numbers?
Answer:
Bob can always determine Alice’s numbers in only two rounds. In the first
round, Bob chooses a1 = a2 = ... = an = 1 and Alice tells him the number
S = x1 + x2 + ... + xn . Note that since all of alice’s numbers are positive, for
all i, xi ≤ S. On the next round, Bob chooses
b1 = 1, b2 =
(S + 1), b3 = (S + 1)2 , ..., bn = (S + 1)n−1 and Alice tells him the number
N = x1 (S + 1) + x2 (S + 1)2 + ... + xn (S + 1)n−1
1