Middle Three Digits
This question was Colombia State University’s Problem of the Week on December 03, 2007. It was also, according to this list, the problem for which the lowest percentage of solutions submitted was correct. (The second “most difficult” problem probably earned that distinction by being poorly specified. And on a second look, I think this one may fit that category as well.) I think that labeling problems as “most difficult” based on the percentage of correct answers is a poor way of making that determination (since for some questions, it will be obvious when an answer is wrong, and the person just won’t bother submitting). I’d think the absolute number of correct answers would be a better guide. Anyhow, the problem:
A five-digit perfect square in the form of 5abc6 has a thousands digit a, hundreds digit b, and tens digit c. If a is less than or equal to b and b is less than or equal to c, what is the sum of a + b + c?
My challenge to you, my readers, is this: is there a better way to solve this than “guess and check”? I see ways to narrow the guessing down substantially, so this may indeed be the best way, but I wonder if there is another approach.
And I’ll also add a combinatorics question to this: How many five-digit numbers are there of the specified form? (And if you like, you could add an answer to the question “Is this a Permutations question or a Combinations question?“)