A calculus of the absurd
2.4 Psychological tactics
The landlady hurried into the backyard, put the mousetrap on the ground (it was an old-fashioned trap, a cage with a trapdoor) and called to her daughter to fetch the cat. The mouse in the trap seemed to understand the gist of these proceedings; he raced frantically in his cage, threw himself violently against the bars, now on this side and then on the other, and in the last moment he succeeded in squeezing himself through and disappeared in the neighbour’s field. There must have been on that side one slightly wider opening between the bars of the mousetrap... I silently congratulated the mouse. He solved a great problem, and gave a great example.
That is the way to solve problems. We must try and try again until eventually we recognize the slight difference between the various openings on which everything depends. We must vary our trials so that we may explore all sides of the problem. Indeed, we cannot know in advance on which side is the only practicable opening where we can squeeze through.
The fundamental method of mice and men is the same: to try, try again, and to vary the trials so that we do not miss the few favourable possibilities. It is true that men are usually better in solving problems than mice. A man need not throw himself bodily against the obstacle, he can do so mentally; a man can vary his trials more and learn more from the failure of his trials than a mouse.
—George Pólya, mice and men
From “The Art and Craft of Problem Solving” by Paul Zeitz.
• Get Your Hands Dirty: This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to figure out why the pattern you see is happening. It is a well-kept secret that much high-level mathematical research is the result of low-tech “plug and chug” methods. The great Carl Gauss, widely regarded as one of the greatest mathematicians in history..., was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to \(x^2 + y^2 \leqq 90,000\).
• Penultimate Step: Once you know what the desired conclusion is, ask yourself, “What will yield the conclusion in a single step?” Sometimes a penultimate step is “obvious,” once you start looking for one. And the more experienced you are, the more obvious the steps are. For example, suppose that \(A\) and \(B\) are weird, ugly expressions that seem to have no connection, yet you must show that \(A=B\). One penultimate step would be to argue separately that \(A \geqq B\) AND \(B\geqq A\).
Perhaps you want to show instead that \(A \ne B\). A penultimate step would be to show that \(A\) is always even, while \(B\) is always odd. Always spend some time thinking very explicitly about possible penultimate steps. Of course, sometimes, the search for a penultimate step fails, and sometimes it helps one instead to plan a proof strategy.
• Wishful Thinking and Make It Easier: These strategies combine psychology and mathematics to help break initial impasses in your work. Ask yourself, “What is it about the problem that makes it hard?” Then, make the difficulty disappear! You may not be able to do this legally, but who cares? Temporarily avoiding the hard part of a problem will allow you to make progress and may shed light on the difficulties. For example, if the problem involves big, ugly numbers, make them small and pretty. If a problem involves complicated algebraic fractions or radicals, try looking at a similar problem without such terms. At best, pretending that the difficulty isn’t there will lead to a bold solution. At worst, you will be forced to focus on the key difficulty of your problem, and possibly formulate an intermediate question, whose answer will help you with the problem at hand. And eliminating the hard part of a problem, even temporarily, will allow you to have some fun and raise your confidence. If you cannot solve the problem as written, at least you can make progress with its easier cousin!