MathDB

k

is a positive integer,

R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}

for

n=2k

R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}

for

n=2k+1

. A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from

R_{n}

with the property that any two connected marbles have different numbers on them, we call it nice labeling. If each one of the marbles are written on some numbers from

R_{n}

with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to

0

, we call it precise labeling.

n\geq{3}

, if every mechanism that is labeled nicely with

R_{n}

, could be labeled precisely with

R_{m}

, what is the minimal value of

m

Problem Statement