Deriving mass/energy equivalence with a simple thought experiment

Anyone who's done any mathematics will remember how some complex problems become trivial when you change the coordinate systems. It's a stunning thing to see and feel. The right framing of a problem can make the solution self evident. We see a similar phenomenon in studies of cognition; a seemingly complex problem can become easy for most people to solve when its translated into a social relationship structure. In special relativity problems changing the 'frame of reference' can make a difficult question easy.

I'm fascinated by these kinds of transformation, though I lack the genius necessary to invent new transformations. I think this way of leading just about any thoughtful person to an understanding of the necessity of mass/energy equivalence is a kind of coordinate system transform:

Why Does E=mc2? | Cosmic Variance

...Now let’s think about a second thought experiment, which is closely related to the first. All I want to make different is to replace the cannon by a powerful laser. Instead of a cannonball being propelled across the box, we’ll now think about the laser firing a pulse of light. Now, the light carries momentum, and so when the laser fires and the pulse sets off, the box will once again begin a backwards slide in order that momentum be conserved. Also once again, when the light reaches the other side and is absorbed by the opposite wall, the momentum will be transferred back to the box, which will then come to a halt. But now you see the problem. The distribution of mass in the box is the same as it was at the beginning, and no external forces have acted on the system, and yet because the box has slid backwards and no mass has been moved, the center of mass of the entire system has moved! All no longer seems right with the world...