Hi Smile,
Thanks for the question! As is often the case with more advanced inferences, this one is the result of a combination of different things. These three rules all play a role here:
- Rule #1: At least one of any three consecutively numbered lights is off.
Rule #4: At least one of the three lights on each side is on.
Rule #5: If any side has exactly one of its three lights on, then that light is its center light.
So, with those in mind, let's go back to your question. Let's start by considering what occurs when one of the corners is off, and let's use just bottom lights 7-6-5 for the discussion (we'll isolate one side entirely so there's no additional confusion from having to consider other lights and rules). What happens if we turn light 5
off?
First, applying Rule #4, we know that either light 6 or 7 must be
on. So, let's now consider what happens if one of those or both are on:
- Light 6 on, Light 7 off: this conforms to all rules (in particular rule #5), and so this is ok.
Light 6 off, Light 7 on: In this scenario, only Light 7 is on, which violates rule #5. so, this is impossible.
Light 6 on, Light 7 on: By itself, within this group of three lights this would be ok (it ultimate violates Rule #1 when Light 8 is thrown in the mix, but we'll ignore that for a moment).
So, you can see that in the only two instances that are acceptable above, the center light is on each time. The one instance where the center light is off, then you can't create an acceptable scenario. Essentially, when you turn one of the corners off, you still have to meet the rule that one light on a side is on. But, that single light can't be the other corner, and so no matter what you do, you have to turn the center on.
Please let me know if that helps clear up this tricky inference. Thanks!
