Cult of Dan Brown: A game of riddles...

With mathematics, like so many other human pursuits, it's more often the journey than the destination that interests us. In fact, if you'd been able to give Archimedes the 206,545-digit answer to his problem, he'd have had no way to verify its correctness. He had no idea what the "correct" answer was, but he knew the algebraic formulas that one would have to use to compute it. To impress Archimedes in his own day and age, you'd have to demonstrate your problem-solving skills--not by arriving at the ultimate solution, but by proving that your methods are guaranteed to do so. If your mathematical reasoning is sound, it follows that you're "on the correct road" to the solution, and that's what's important.

The Drexel website illustrates an approach to solving the problem using modern mathematical tools--tools that Archimedes didn't have, like matrices and linear algebra. These certainly help express the solution more compactly, but they're hardly necessary. Simple algebra is sufficient, even if it requires more tedious, long-hand math to solve a system of seven equations with recursive substitution.

Archimedes would have wanted to see two things from a student:

(1) A proof that a solution exists;

(2) A logical roadmap to that solution.

Proving that a solution exists is really the harder part, given that Archimedes has imposed so many conditions on the problem (nine in total). In general, there's no guarantee that a system of equations will have a solution at all--it may be impossible to satisfy all of the conditions at the same time. A system of equations may have 0, 1, or an infinite number of solutions, and in this particular problem the latter is the case. By the time you work out that:

102,571,605,819,606r^2 = m(m + 1)

you know that at least one solution exists for some integer value of r and some integer value of m. The 206,545-digit result is merely the smallest of these solutions; there exists an infinite number of larger solutions for different combinations of r and m. Solving for r and m, however, requires trial and error, hence brute calculating force that wasn't available in Archimedes' day. He'd have to have been satisfied with the proof that a solution exists, in other words, since an actual solution was practically unobtainable. A smartass student could technically have answered that the number of cattle is infinite--but he'd have needed series calculus to prove it

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