Eight light bulbs are placed on the eight lattice points $$( \pm 1, \pm 1, \pm 1)$$. Each light bulb can either be turned on or off. However, the lightbulbs are unstable, and if two light bulbs with distance less than or equal to $$2$$ are on simultaneously, both lights explode. How many possible configurations of on/off light bulbs exist if no explosions occur?

This problem is shared by Muhammad A. Lewis Chen proposed it for an NIMO competition.

Details and assumptions

The 8 positions of the light bulbs correspond to the cases when each of the lattice points are either $$+1$$ or $$-1$$. This gives a total of $$2 \times 2 \times 2 = 8$$ possible positions for the light bulbs.

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