A million years ago I had a professor named Al Andry. I encountered him in a mid-Western airport one Friday as I was returning from a job interview and he was jetting off to a conference somewhere.
We chatted socially for a little while even though I was a student and he was a prof.
He shared his theory of testing. For one thing, he thought that the test lacked resolution if everybody passed, that is, if all the students scored between 70% and 100%. In his mind, he had written a great test if some students scored 5% and others scored 95%. That would make him an unpopular professor today.
His tests always had four problems. Each problem was worth 25%. Two were exactly like the homework. Those problems were floaters drifting down the center of the strike zone. You were virtually guaranteed at least 50% on his tests if you simply did all of the assigned homework and understood it.
One question was "fringes of knowledge". You had to be diligent to get any traction on the third problem.
The final kind of question was sneaky. Diligence often worked against you on this kind of problem. You needed perspective to make headway on the fourth kind of problem.
One of Professor Andry's sneaky questions involved multiplying three, massive arrays. It may have been finding the kinetic energy of system when the velocities were expressed as three-component vectors. The formula for something like that would be something like 0.5w^2(U^t)[M](U) where U (the displacement) was a 7X3 array and the M was a 7X7 matrix. It works out to something like 170 separate multiplication operations and a bunch more addition. At any rate, the number of computations was staggering for a college kid with a TI-30 and fifteen minutes left on the clock.
Andry was a bit disgusted when he handed back the tests.
The key, he said, was to LOOK at the problem. Look at the mass matrix. Look at the U matrix. There were lots, and lots of zeros. In fact, if you thought to look for a pattern you would quickly see that there was only one path where the "answer" did not get cock-blocked by a zero somewhere.
And, if I remember correctly, the answer was 1.0*1.0*1.0, an answer easily found by inspection.
I don't know if he intended for this to be a life lesson although it turned out to be good advice. Don't keep pushing against stones that won't move. Don't waste love on people who it bounces off of. Don't put emotion into events you cannot influence or prepare for. Don't invest in stale cigars.