1. Existence proof

Claim: object with given properties exists, or not exists.
Proof: an example

For example, if you claim at least one swan is white, you just need to find a white swan.
Another example, if someone claims that all swans are white, you just need to find a black swan to disapprove it.
Notice that finding a white swan will not prove this statement.

2. Invariants

The properties that don’t change are called invariants. They maybe numbers, formula, trend…

It’s always a good idea to first find invariants before jumping straight into the problem. They will provide key insight on the problems.

For example, how to solve magic square? For 3X3 ones, sum from 1 to 9 is 45, thus for single line is 15. From this we can conclude that the number in the center is 5. This is an invariant. Knowing this fact will help us narrow the total search space.

3. Reuse knowledge

We can transform known solutions into solutions for new problems. The whole edifice of computer science is built on this. Reusable codes are just some powerful demonstrations.

The hard point is how to find solved problems in a new given problem. This needs insight and flash of creation. A common theme is divide&conquer.

Here’s a joke about mathematician. A man asks a mathematician a question ”what will you do if you walk into a street and find a building on fire?”. “First I’ll call fire department. Second I’ll find a bucket and water. Third I’ll try to put out the fire.” Answered the mathematician. “What will you do if you walk into a street and find everything OK?” “I’ll set a fire.” “Why will you do that!?” said the man. “Because this will transform the current question into a question I’ve already solved.” Answered the mathematician.

4. Induction

The core concept of induction is assuming we already proved a statement, if we can push the statement a little further, we finish proving that statement.

General form:
Induction base: prove an easy case
Induction step: from n to n+1

Example: Bernoulli’s Inequality
Theorem: for any n>=0, x>0, we have (1+x)ⁿ >= 1+nx
Proof by induction:
Base case: when n = 1, (1+x)ⁿ = (1+x)⁰ = 1 = 1+0x = 1 + nx
Step: (1+x)ⁿ⁺¹ = (1+x)ⁿ(1+x) >= (1+nx)(1+x) = 1 + (n+1)x

Another variation of induction is complete induction. In complete induction we assume all previous cases are true, while in weak induction we only assume the case n is true.

5. Reductio ad Absurdum

The core concept is proof by contradiction.

General form:
Assume the conclusion is true. Proceed to reach a contradiction, usually contrary to problem restriction or common sense.

For a famous example check here.
https://en.wikipedia.org/wiki/Mathematical_proof#Proof_by_contradiction.