Let
be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses n real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are
such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial n numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a.
b.
c.![n=8](/media/m/2/6/1/261135591c44485b829036fc3749c0b6.png)
Justify your answer(s).
[For example, when n=4, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
- Alice chooses n real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are
![\frac{n(n-1)}{2}](/media/m/6/3/4/6349f59c04c06e2a8420d43c7533f79c.png)
- Bob wins if he finds correctly the initial n numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a.
![n=5](/media/m/3/b/2/3b24f190cdbea6325c7013bc196e0838.png)
b.
![n=6](/media/m/4/1/2/412ce8d153155e39339fdb2acbc64e69.png)
c.
![n=8](/media/m/2/6/1/261135591c44485b829036fc3749c0b6.png)
Justify your answer(s).
[For example, when n=4, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]