The grandfather of Baron K.F.I. von Munchausen built a square castle, divided it into 9 square halls, and placed an arsenal in the central hall. The father of the baron divided each of the eight remaining halls into 9 equal square foyers and arranged winter gardens in all the central foyers. The baron himself divided each of the 64 free foyers into 9 equal square rooms and set up a pool in each of the central rooms, making the rest living rooms. The baron boasts that he managed to visit all the living rooms, visiting each one only once, and returned to the starting point (a door was made in each wall between two adjacent living rooms). Could the baron's words be true?
This one wants a proof. Work it on paper, read the official solution, then mark
yourself honestly — the ladder only means something if the record is true.
Official solution
First, let's walk around one hall, for example, clockwise (left figure). Consider the adjacent hall and walk around it similarly. To walk around both halls, it is sufficient to change the direction of movement in two pairs of cells along the boundary of the halls (center figure). Consider the next hall that borders the already walked around ones. By changing the direction of movement similarly to the previous one, we get a way to walk around three halls (right figure). By adding new halls one by one and changing the direction of movement in the specified manner, we will be able to walk around the entire castle. !
## Answer
They can.
## Problem
Source: NuminaMath-1.5,
licensed Apache-2.0.
Statement and solution reproduced as published; topic, difficulty and ordering added
by this site.