MathDB

A ten-level

2

-tree is drawn in the plane: a vertex

A_1

is marked, it is connected by segments with two vertices

B_1

and

B_2

, each of

B_1

and

B_2

is connected by segments with two of the four vertices

C_1, C_2, C_3, C_4

(each

C_i

is connected with one

B_j

exactly); and so on, up to

512

vertices

J_1, \ldots, J_{512}

. Each of the vertices

J_1, \ldots, J_{512}

is coloured blue or golden. Consider all permutations

f

of the vertices of this tree, such that (i) if

X

and

Y

are connected with a segment, then so are

f(X)

and

f(Y)

, and (ii) if

X

is coloured, then

f(X)

has the same colour. Find the maximum

M

such that there are at least

M

permutations with these properties, regardless of the colouring.

Problem Statement