National problem · Combinatorics

Timotej had a sheet of squared paper measuring \(8 \times 8\) little squares. He folded it a few times, each fold running along one of the lines of the grid, until he was left with a square piece measuring \(3 \times 3\) little squares. He put this folded piece on the table and wrote in each of its \(9\) squares how many layers of paper that square contains. The picture shows one possible sequence of folds together with the record of layer counts it produces.

One possible sequence of folds. Each dashed line is a fold line, and the smaller part of the sheet is turned over onto the rest.

(a) Find a sequence of folds for which Timotej would obtain the record \(A\).

(b) Prove that Timotej could not have obtained the record \(B\).

The three records \(A\), \(B\) and \(C\).