Muhammad's Light Lattice

Discrete Mathematics Level 5

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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