The median score for college-bound seniors on the math section of the SAT in 2011 is about 510 out of 800. So right there is proof that there are lots of unsolved math problems.

The great 19th-century mathematician Carl Friedrich Gauss called his field “the queen of sciences.”

If math is a queen, she’s the White Queen from Alice in Wonderland, who bragged that she believed “as many as six impossible things before breakfast.” (No surprise that Lewis Carroll also wrote about plane algebraic geometry.)

For example, the Navier-Stokes equations are used all the time to approximate turbulent fluid flows around aircraft and in the bloodstream, but the math behind them still isn’t understood.

And the oddest bits of math often turn out to be useful. Quaternions, which can describe the rotation of 3-D objects, were discovered in 1843. They were considered beautiful but useless until 1985, when computer scientists applied them to rendering digital animation.

Some math problems are designed to be confounding, like British philosopher Bertrand Russell’s paradoxical “set of all sets that are not members of themselves.” If Russell’s set is not a member of itself, then by definition it is a member of itself.

Russell was using a mathematical argument to test the outer limits of logic (and sanity).

Kurt Gödel, the renowned Austrian logician, made matters worse in 1931 with his first incompleteness theorem, which said that any sufficiently powerful math system must contain statements that are true but unprovable. Gödel starved himself to death in 1978.

Yet problem solvers soldier on. They struggled for 358 years with Fermat’s last theorem, a notoriously unfinished note that 17th-century mathematician and politician Pierre de Fermat scrawled into the margin of a book.

You know how 32 + 42 = 52? Fermat claimed that there are no numbers that fit the pattern (an + bn = cn) when they are raised to a power higher than 2.

Finally, in 1995, English mathematician Andrew Wiles proved Fermat was right, but to do it he had to use math Fermat never knew existed. The introduction to Wiles’s 109-page proof also cites dozens of colleagues, living and dead, on whose shoulders he stood.

At a conference in Paris in 1900, German mathematician David Hilbert determined to clear up some lingering math mysteries by setting out 23 key problems. By 2000 mathematicians had solved all of the well-formed Hilbert problems save one–a hypothesis posed in 1859 by Bernhard Riemann.

The Riemann hypothesis is now regarded as the most significant unsolved problem in mathematics. It claims there is a hidden pattern to the distribution of prime numbers—numbers that can’t be factored, such as 5, 7, 41, and, oh, 1,000,033.

The hypothesis has been shown experimentally to hold for the first 100 billion cases, which would be proof enough for an accountant or even a physicist. But not for a mathematician.

In 2000 the Clay Mathematics Institute announced $1 million prizes for solutions to seven vexing “Millennium Prize Problems.” Ten years later the institute made its first award to Russian Grigori Perelman for solving the Poincaré conjecture, a problem dating back to 1904.

Proving that mathematicians don’t grasp seven-digit numbers, Perelman turned down the million bucks because he felt another mathematician was equally deserving. He currently lives in seclusion in Russia.

In his teens, Evariste Galois invented an entirely new branch of math, called group theory, to prove that “the quintic”—an equation with a term of x5—was not solvable by any formula.

Galois died in Paris in 1832 at age 20, shot in a duel over a woman. Anticipating his loss, he spent his last night frantically making corrections and additions to his math papers.

Graduate student George Dantzig arrived late to statistics class at Berkeley one day in 1939 and copied two problems off the blackboard. He handed in the answers a few days later, apologizing that they were harder than usual.

The “homework” was actually two well-known unproven theorems. Dantzig’s story became famous and inspired a scene from Good Will Hunting.

Peter Coy is economics editor of Bloomberg Businessweek.