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## Seven Jack O'Lanterns

Logic puzzles require you to think. You will have to be logical in your reasoning.

 Puzzle ID: #20639 Fun: (3.04) Difficulty: (2.48) Category: Logic Submitted By: fishmed Corrected By: boodler

It appears that you have angered the spirit of Halloween by failing to revere the Great Pumpkin, and now a curse has befallen you. On the walkway to your house is a Ward of Seven Jack O'Lanterns arranged in a circle. If midnight comes and any of the seven are still lit, a dark reaper and seven dark horses with seven dark riders shall visit thy abode. They shall surround thy domicile and, while circling it, they will proceed to pelt thy dwelling with eggs and cream of shaving. And come morn there will be a great mess to be reckoned with. Verily. So you better get those lanterns out.

You quickly discover something odd about these lanterns. When you blow out the first one, the lanterns on either side extinguish as well! But there is more. If you blow out a lantern adjacent to one that is extinguished, the extinguished one(s) will relight. It seems that blowing on any lantern will change the state of three - the one you blew on and its two neighbors. Finally, you can blow on an extinguished lantern and it will relight, and its neighbors will light/extinguish as applicable. After trying once and finding all seven lit again, you decide, being the excellent puzzler, you sit down and examine this closer. But hurry, I hear the beating of many hooves...

If you examine the setup carefully, you'll note a number of facts which make the puzzle easier to solve by deduction. First, blowing on a lantern is a commutative property; blowing on lanterns 1, 5, then 3 is the same as blowing on 3, then 1, and then 5. No matter what order the lanterns are blown on, if the same lanterns are blown on the same number of times, the result won't change. For that reason, blowing on a lantern twice is as good as not blowing on it at all. And three times is as good as one time. So, it seems that it should be able to be done in seven steps or less.

What else can we tell about the solution? Since each operation changes the state of three lanterns, and there are 7 lanterns, and each lantern must change its state an odd number of times, it's a safe bet that there will need to be an odd number of steps. We can easily see it can't be done in 1 or 3 steps, so it must be 5 or 7. Trying 5 steps comes up with 3 different patterns that are not symmetrical and fail to leave all lanterns extinguished. So that leaves 7 steps and to your surprise, based on the commutative property, the easiest solution is to blow on each one in order! So doing this, the Great Pumpkin has decided to give you a treat for figuring this out and you find all seven lanterns full of candy the next morning! Congratulations!

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