"There are a number of accepted “styles” of doing proofs. Here are some important ones: Direct proof. The examples we’ve used up to now have been direct proofs. This is where you start from what’s known and proceed directly by positive steps towards your conclusion. Direct proofs remind me of a game called “word ladders,” invented by Lewis Carroll, that you might have played as a child: WARM |||| ???? |||| COLD. You start with one word, like WARM, and you have to come up with a sequence of words, each of which differs from the previous by only one letter, such that you eventually reach the ending word, like COLD. It’s sort of like feeling around in the dark: WARM WART WALT WILT WILD |||| .... This attempt seemed promising at first, but now it looks like it’s going nowhere. (“WOLD?” “CILD?” Hmm....) After starting over and playing around with it for a while, you might stumble upon: WARM WORM WORD CORD COLD. This turned out to be a pretty direct path: for each step, the letter we changed was exactly what we needed it to be for the target word COLD. Sometimes, though, you have to meander away from the target a little bit to find a solution, like going from BLACK to WHITE: BLACK CLACK CRACK TRACK TRICK TRICE TRITE WRITE WHITE. Here, we had to temporarily change our first letter three different times—two of which seemingly brought us no nearer to WHITE—in order to successfully forge a path through the tangled forest. Knowing which direction to set out on is a matter of intuition plus trial and error. Given the axioms of any system, whether algebra, predicate logic, sets, etc., there are an unfathomable number of different ways to proceed. The vast majority of them are bound to lead to dead ends. This is why a valid proof, when it is finished, is often an elegant and beautiful thing. It’s a thin braid of jewels glistening in the midst of a whole lot of mud."