On the Domino Ring

Probability Level 3

Consider a set of N(N+1)2 \frac{ N (N+1) } { 2 } dominos which have faces of values 1 to N.
A ring of dominos is a closed loop of them, in which any 2 touching dominos display the same value.
Let F(N)F(N) be the minimum numbers of rings that are needed to use up all of these dominos.

Find the last three digits of n=12016F(n)\displaystyle \sum _{n = 1} ^{2016} F(n) .


  • As an explicit example, F(3)=1 F(3) = 1 because we have the ring listed in the above image.
  • For the purposes of this question, assume a single domino to be a ring. So F(2)=3 F(2) = 3 (see image below) because there are three rings.


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