MathDB

Let

n

be a positive integer. Let

B_n

be the set of all binary strings of length

n

. For a binary string s_1\hdots s_n, we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by

b

. Then, we replace

s_b

with

1-s_b

. A string

a

is said to be a descendant of

b

a

can be obtained from

b

through a finite number of twists. A subset of

B_n

is called divided if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of

B_n

.
Remark. Here is an example of a twist:

101100 \rightarrow 101000

because

1\mid 0\mid 11\mid 00

has

4

blocks of consecutive digits.
Viktor Simjanoski

Problem Statement