George Polya: Let's teach guessing

all-important reminder that we must test our guesses
Guess and then prove. All great discoveries are done this way
Finished math consists of proofs, but math in the making/discovery consists of guesses

Students started with random guesses
Start with easier problems which could help
A student asked if planes can be parallel, and prompted the clarification that planes are random
3 planes give 8
Think of extreme cases in reasonable guessing
Guess 4 planes give 16 based on observation of pattern, and then generalize, i.e., induction
Need to test the guess/prediction
Use analogy to help guess, what if lines against planes? Turns out 4 lines confine a triangle, and so do 4 plane
Turns out 3 lines give only 7, which suggests that plane version is incorrect. We could but 1 triangle + 3 side adjecent + 3 vertex adjecent. In plane case, it is 1 finite + 4 faces + 6 cutting edge + 4 points
Analogy: line divided by points, and then we guessed by induction when we see the pattern with lower dimensions
Prove the pattern with 5 case is hard, so we try 4 + line case. After we verified the 11 case, we are more confident with the 26 guess.
Proving the answer is out of scope for this lecture