Geometry – Question 11 of Article III of the BMI (International Mathematics Contest) of 2011 on the evaluation of an integer k

A checkerboard is placed on a square of an infinite checkerboard, where each square measures 1 cm by 1 cm. It moves according to the following rules:

1 / At the first stroke, the pawn moves one block to the north

2 / All odd movements are north or south and all even movements are east or west

3 / At the nth displacement, the controller moves n squares in the same direction.

The auditor makes 12 moves so that the distance between the centers of his initial and final squares is as small as possible. What is this minimum distance?

For the above question, I thought that the shortest distance would be determined if the verifier moved in a spiral. Unfortunately, that does not give the right answer.

Can you, please, inform me, what kind of form I should have thought to use to achieve the minimum distance, and with this form, how far does it seem to be ?

thanking you in advance

Kevin