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This is a Circular Linearity, Identify the Possibilities game.
The first rule establishes that at least one of any three consecutively numbered lights is off, meaning three lights in a row cannot be on:
The second rule establishes that light 8 is on:
The third rule states that lights 2 and 7 cannot be on when light 1 is on:
1 2, 7
This rule will play a pivotal role in an inference to be discussed shortly.
The fourth rule indicates that at least one of the three lights on each side is on:
The fifth rule is another rule about sides and lights, and it indicates that if exactly one light on a side is on, then that light must be the center light:
Side 1 light on Center on
The contrapositive of this rule is:
Center on Side 1 light on
Since a side must have at least one light on and cannot have all three lights on, this contrapositive can be translated as:
Center off Side 2 lights on
When a side has two lights on but the center is not on, then both corners must be on:
Center off Both corners on that side are on
The contrapositive of this inference is:
Both corners on that side are on Center on
Thus, if one of the corners is off, then the center light is automatically on.
The final rule states that two lights on the north side are on. From the third rule we know that lights 1 and 2 cannot be on at the same time, so, by Hurdling the Uncertainty we can infer that light 3 must always be on (otherwise you could not fulfill the constraints of this rule):
At this point, most students move on to the questions. But, there are six rules, and several of those rules establish general limitations on each side or section of three lights, and these rules, when combined with the fact that the status of two of the eight lights is already determined, indicate that the game cannot have a large number of solutions. The best decision, then, is to explore Identifying the Possibilities.
Start first with the third rule, which states that lights 2 and 7 are off when light 1 is on. By turning light 1 on, lights 2 and 7 automatically are off, leaving lights 4, 5, and 6 undetermined. But, from our discussion of the fifth rule, when a corner light is off (as light 7 is), then the center light on that side is on. Hence, light 6 must be on. Lights 4 and 5 cannot be precisely determined, but if one is on, the other is off (if both were on, the first rule would be violated), leading to a dual-option. Combining all of the information gives us only two possibilities when light 1 is on:
Of course, light 1 could be off. In that case, light 2 must be on in order to meet the constraints of the final rule. With lights 2 and 3 on, light 4 must be off in order to conform to the first rule. With light 4 off, light 5 must be on in order to abide by the fifth rule. The only undetermined lights are 6 and 7, but both cannot be on (otherwise the first rule would be violated) and both cannot be off (otherwise the fifth rule would be violated). Thus, one of lights 6 and 7 is on, and the other is off, leading to two possibilities:
Thus, because all possibilities have been explored when light 1 is on and when it is off, and light 1 has no more possible positions, we have explored all possibilities of the game, and there are only four possible solutions, as captured by the two templates above.
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