16
Explanation
For every triangle whose sides are a, b, and c, these three inequalities
must hold:
a+b < c
a+c < b
b+c < a
So every triangle whose sides are 14, 3, and c, these three inequalities
must hold:
14+3 > c
14+c > 3
3+c > 14
These become
17 > c
c > -11 <--that will always hold!
c > 11
So 11 < c < 16
So the only values for the third side are between 11 and 17, exclusive.
That means the third side is either 13, 14, 15, or 16.
The largest possible whole number length is 16.
