Will is given 10 rods, whose lengths are all distinct integers. However, he finds that given any 3 rods, he is unable to construct a (non-degenerate) triangle with them. What is the shortest possible length for the longest rod?
Details and assumptions
A non-degenerate triangle is a triangle with positive area.
is the answer 55
no
To find the shortest possible length for the longest rod, let's consider the given condition that no three rods can form a non-degenerate triangle.
In a triangle, the sum of the lengths of any two sides must always be greater than the length of the third side. However, if we cannot form a non-degenerate triangle with any three rods, it means that the sum of the lengths of the two shorter rods will always be less than or equal to the length of the longest rod.
To minimize the length of the longest rod, we need to maximize the sum of the lengths of the two shorter rods. Since all 10 rod lengths are distinct integers, we should choose the two shortest rods.
Since the lengths are distinct integers, we can assume the smallest two rods have lengths 1 and 2. Therefore, the sum of the lengths of these two rods is 1 + 2 = 3.
To find the shortest possible length for the longest rod, we need to find the maximum value less than or equal to 3 that is not present among the given rod lengths.
Let's assume the longest rod has length 'L'. To satisfy the condition, 'L' should be less than or equal to 3 and not present among the given rod lengths.
We can start checking the values in decreasing order: L = 3, 2, 1. If any of these values are not present among the rod lengths, it can be the shortest possible length for the longest rod.
Note: Be sure to check all the given rod lengths to ensure that none of them match the condition.
In this case, as 3 is present among the given rod lengths, we continue checking.
Checking for L = 2:
If 2 is not present among the given rod lengths, then 2 is the shortest possible length for the longest rod. However, if 2 is present, we continue checking.
Checking for L = 1:
If 1 is not present among the given rod lengths, then 1 is the shortest possible length for the longest rod. Otherwise, we need to check the next smaller value, but in this case, as the range of possible values is exhausted, the shortest possible length for the longest rod is not found.
Therefore, the given information does not provide enough information to determine the shortest possible length for the longest rod.