MathDB

A board

1

k

is called guayaco if: -Each unit square is painted with exactly one of

k

available colors. -If

gcd(i,k)>1

, the

i

th unit square is painted with the same color as

(i-1)

th unit square. -If

gcd(i, k)=1

, the

i

th unit square is painted with the same color as

(k-i)

th unit square. Sebastian chooses a positive integer

a

and calculates the number of boards

1

a

that are guayacos. After that, David chooses a positive integer

b

and calculates the number of boards

1

b

that are guayacos. David wins if the number of boards

1

a

that are guayacos is the same as the number of boards

1

b

that are guayacos, otherwise, Sebastian wins. Find all the pairs

(a,b)

such that, with those numbers, David wins.

Problem Statement