Meng also gave a less intricate constructive proof.

Since then several constructive proofs have been found.

A short constructive proof was presented in :

Constructivism is a mathematical philosophy that rejects all but constructive proofs in mathematics.

A constructive proof of the theorem would give an actual example, such as:

Such counterexamples do not disprove a statement, however; they only show that, at present, the statement has no constructive proof.

Because no such proof is known, the quoted statement must also not have a known constructive proof.

In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it.

Also, it is not a constructive proof: it does not exhibit a concrete position that needs this many moves.

The constructive proof applies it to the sequence of real algebraic numbers, thus producing a transcendental number.