### Brain Teasers

# Seven Jack O'Lanterns

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...

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...

### Answer

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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## Comments

Very nice teaser. I have seen something similar done with a wall of bricks where the four bricks surrounding the first brick become recessed.....FUN!

This was hard to me because i have a short attention span and ummm what was i talking about?

nice one but is a bit long

This is one of my favorites. I didn't really look at it the way you did though, I just figured out what pattern they had to be in to extinguish them all, and then I got to that point. It is interesting though because it took me exactly seven steps. Nice one!

Actually it can be done in seven steps

Imagine these are the seven (B=burning, O=Out)

Start: BBBBBBB

#1: OOOBBBB

#2: OOBOBBB

#3: OOBBOBB

#4: OOBBBOO

#5: OOOOOOO

Imagine these are the seven (B=burning, O=Out)

Start: BBBBBBB

#1: OOOBBBB

#2: OOBOBBB

#3: OOBBOBB

#4: OOBBBOO

#5: OOOOOOO

whups, I meant 5 steps

again whups i did it wrong I thought i had checked it, but i guess i didn't can somebody delete these three posts?

I just requested the last 3 were removed for you. I am glad you guys liked it.

u add way to much information... just tell me wat i need to know and get it over with... i get lost when there is too much info

Very well written and entertaining.

Fabulous Job!!!

Fabulous Job!!!

Thanks. Glad you liked it.

Nothing wrong with a good storyline.

loved this one! i guess i'm one of few who likes the funny story line?

took me a little while to figure out. nice work.

took me a little while to figure out. nice work.

took me a little while to figure out. nice work.

Like Peanuts?

Of course I do!

great! long! but one of the bests

Thanks.

too long... i got lost on the first paragraph

heh. I blew out the two corner ones and then the middle one

I loved the story and the challenge

rotwyla, they are in a circle, so there are no 'corner' ones. Nice thought, though.

I like this teaser, but I loved the answer. Explaining it in with the commutative power is wonderful. I did answer it, but with trial and error and I did not come close to understanding the mathmatical proof for solving. (actually the answer was the first attempt I made)

WHAT ???

yah ... thanks alot ! now i'm scared of jack-o-lanterns !

yah ... thanks alot ! now i'm scared of jack-o-lanterns !

Nicely done, Fish! Kudos!

I'm really lost, who or what is Jack O'Lanterns? But seems like a well liked teasers.

Jack O' Lanterns are hollowed-out pumpkins that have faces carved into them and then are lit from inside with a candle. Usually used for decorations during Halloween.

Nice ripoff, this was originally posted at www.greylabyrinth.com back in 1998.

I actually got it from my father. I am not sure where he got it from.

this one was cleverly nice

nice one to the good teaser

Fun teaser but quite difficult!

This was funny, entertaining and unique... I loved it!

I cut 7 little squares from paper, drew a little lit candle on the one side and left the other blank then arranged them in a circle, lit side up. When I altered one, I flipped the two adjacent squares. Wasn't too difficult. Trick was to "blow out" all but one then blow out the two middles of the lit ones.

Anyway, a lot of fun!

I cut 7 little squares from paper, drew a little lit candle on the one side and left the other blank then arranged them in a circle, lit side up. When I altered one, I flipped the two adjacent squares. Wasn't too difficult. Trick was to "blow out" all but one then blow out the two middles of the lit ones.

Anyway, a lot of fun!

Excellent teaser! I don't generally like too much extraneous story, but I thought it was amusing enough and not too long, so it was fine.

I missed the commutive property, but came up with another line of reasoning to determine that seven steps were the minimum.

Each step toggles three candles. You need an odd multiple of seven toggles in order to toggle the seven candles off. The least common multiple between three and seven is 3 x 7 = 21, so seven steps is the minimum. Then I just guessed that you could blow them out in order and was mildly surprised when is worked!

I missed the commutive property, but came up with another line of reasoning to determine that seven steps were the minimum.

Each step toggles three candles. You need an odd multiple of seven toggles in order to toggle the seven candles off. The least common multiple between three and seven is 3 x 7 = 21, so seven steps is the minimum. Then I just guessed that you could blow them out in order and was mildly surprised when is worked!

Feb 20, 2010

It is easier than you describe. All lanterns need to be switched an odd number of times, so the total number of switches must fall in the sequence 7, 21, 35, 49 etc. but since 3 are switched with each blow the total must also be divisible by 3. The first such number in the sequence is 21, meaning 7 blows are required (7x3=21). By blowing a lantern once and the lanterns either side once as well the lantern in question ends up being blown out. It is clear from this that each must be blown once because all are blown once, and for each lantern both either side are blown once - meaning they all end up being out.

I didn't really analyze very much. Just tried it out and got it on the first try with 7 blows. Reading the answer and finding out that the order of the blows didn't really matter made me laugh... Great teaser!

I did it in seven steps, but not in the same way the solution gives. XD

Wow!

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