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## 'Impossible Triangles?' printed from http://nrich.maths.org/

You are given a limitless supply of triangular jigsaw pieces of
type T1, T2, T3, T4 and T5

Using these pieces, you need to try to make larger triangular
shapes without any overlap.

Three triangle shapes are shown in the picture below. Is it
possible to create two-coloured triangles of these shapes without
moving the pieces already placed?

Suppose next that you can move the pieces and choose two triangle
types to work with. Which pairs of shapes can be used to make
larger equilateral triangles? Which pairs of shapes can be used to
make larger 30-60-90 triangles? See the video clip for a discussion
on this part of the problem

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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