NOTES AND BACKGROUND
This problem is all about either finding solutions or proving that there is no solution for any size of triangle shape.
For small triangle shapes, it is easy to check all possible configurations of pieces to check whether a solution exists. For larger triangle shapes the number of combinations of pieces gets larger extremely rapidly, and quickly reaches the point at which a check of all of the combinations is impossible, even on a supercomputer. Even if we have checked a large number of jigsaw sizes and found no
solution, this does not necessarily mean that we cannot find a solution for a larger triangle shape. To find a solution, you often need to mine the depths of your cunning and ingenuity. To show that a solution does not exists you often have to use the concept of proof by contradiction
Proving that a solution does not
exist is often much easier than proving that a solution does exist. Interestingly, if a solution can be found, then it is usually very quick to check that the solution is correct, although finding the solution in the first place might be exceptionally difficult.This behaviour underlies the notion of the mysteriously titled
'P vs NP' problem, the solution of which will earn the solver $1,000,000
Ideas concerning proof are discussed in the fascinating article Proof: A Brief Historical Survey