So the pemise is that you have to write 4 programs, but you can not, they all would fail for some inputs.
1. The first Gödel's incompleteness theorem. A program has to take a set of axioms and a set of true statements about natural numbers. It has to return proofs for all the statements. How it will fail? It would either for some statements return a proof that contradicts itself (so the set of axioms is inconsistent) or return no proof at all (so the set of axioms is incomplete).
2. The second Gödel's incompleteness theorem. A program has to take a set of axioms. I has to return true if program 1 given theese axioms would never retrun a proof that contradicts itself. How it will fail? It would return true for some set of axioms that are inconsistent.
3. Entscheidungsproblem. A program has to take a set of axioms and a statement. It has to return true if the statement is true. How it will fail? It would return true for some false statement.
4. Halting problem. A program has to take a program (in a Turing complete language). It has to return true if the program will ever finish. How will it fail? It would return true for program that would never finish. Or probably would itself never finish. This is also possible for all other programs.