Getting to blue

Probability Level 4

You are on an infinite triangular lattice with blue circles repeated as shown below, and you start at the vertex circled in red:

Every move you randomly walk along a black line segment to a neighboring vertex.

What is the expected value for the number of moves before you hit one of the lattice points with a blue circle on it?

6 5 2 7 3 4 8 1

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Jun 22, 2016

There are two types of "non-blue" lattice points:

  • Those that neighbor a blue lattice point
  • Those that don't neighbor a blue lattice point

Let E 1 E_1 and E 2 E_2 be the expected number of moves to get from each to a blue lattice point respectively.

Then:

  • E 1 = 1 + ( 1 / 2 ) ∗ E 1 + ( 1 / 3 ) ∗ E 2 E_1 = 1 + (1/2)*E_1 + (1/3)*E_2
  • E 2 = 1 + E 1 E_2 = 1 + E_1

Or, E 1 = 8 E_1 = \boxed8

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