What is “Mathematical Alchemy?”

So what exactly is mathematical alchemy? I designed it as a term to capture something that seems to happen a lot in mathematics (both in elementary and research contexts), but doesn’t have a name attached to it. Mathematical alchemy refers to a rephrasing or other transformation of a problem that, while mathematically equivalent to the original formation, encourages the use of a different set of mathematical tools. Even if the mathematical equivalence is not rigorously checked, trying to rephrase a problem with a different terminology can help one’s mind move away from the problem being faced and towards other problems it has already seen and solved.

For instance, one of the basic objects of plane geometry is the angle, and many a good problem is about finding the measurements of some angles given information about other angles (or entire shapes). Trying to write information about angle sizes in the form of equations can encourage someone to solve a problem about angles algebraically. If you know two of the three interior angles of a triangle (call them a_1 and a_2), you can find the measure of the third angle, a_3, with the equation

a_1 + a_2 + a_3 = 180

by solving it for a_3 in terms of the known values of a_1 and a_2.
Of course, this concept of mathematical alchemy can be extended to other, more complicated problems as well, and at the highest levels, this can help break open problems that were previously thought to be impossible to solve. Select mathematicians such as Terence Tao have gained their…