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?
To find the shortest possible length for the longest rod, let's analyze the given information. We know that Will has 10 rods with distinct integer lengths, and any combination of 3 rods cannot form a non-degenerate triangle.
In a non-degenerate triangle, the sum of any two sides must be greater than the length of the third side. So, if we cannot form a non-degenerate triangle with any 3 rods, it means that the sum of the two shortest rods must be less than or equal to the length of the longest rod.
To determine the shortest possible length for the longest rod, we need to find the largest possible sum of the two shortest rods. Let's consider the worst-case scenario where the rods are arranged in increasing order. In this case, the two shortest rods would be the first and second smallest lengths.
Let's assume the smallest rod has a length of 1 unit. Since all the rod lengths are distinct integers, the second-smallest rod must have a minimum length of 2 units. In this scenario, the sum of the two shortest rods is 1 + 2 = 3 units.
So, we can conclude that the shortest possible length for the longest rod is 3 units.