Adults Baffled By Primary School Math Problem - How To Solve The Lighthouse Puzzle

Another day, another viral math problem! Lighthouses A, B, and C turn on and off in regular intervals. A shines light for 3 seconds, then is off for 3 seconds, and repeats this cycle. B cycles 4 seconds on/4 seconds off, and C cycles 5 seconds on/5 seconds off. They all just turned on. What is the first time all three lights will be off? When will they all come on together at the same moment?

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[MindYourDecisions]

Thanks to Insider for featuring this video!

Do subscribe since my videos will make you great at math. Just see how many people find this problem to be easy--obviously my videos are effective so they want harder problems. This is the best proof of how my videos are making people around the world great at math.

[salerio]

Best advice is to never read the Daily Mail, The Sun and especially Mumsnet

[andymcl]

Given that the explanation took about 2mins, it would have been really slick to have the 3 lighthouses in the corner during this and have them sync up at the end of the video :p

[thomasmaughan]

It was pretty easy and I enjoyed noticing the tricky part about when will they all be off at the same time. Not *turning off* but *merely off* and that's after 5 seconds.

[aloox]

The fact that this baffled parents is amazing lol

[mismag]

I thought the first answer was 60 seconds because I DIDN'T READ THE QUESTION and quickly thought all the lights turned off for only 1 second. So A - would turn turn off every 4th sec
B- would turn off every 5th sec
C would turn off every 6th sec
The least common multiple of 4, 5 & 6 would be 60

[AA-]

The problem is that the answer to the 2nd problem is debatable. If you are talking about the *moment* all 3 lights are turned on at the same time the answer is of course 120 sec. Some people interpret the question as asking for the next time *all 3 lights are on* and not *turned on simultaneously* . In this case all 3 lights are on at 24 sec.

And on top of that people can also misinterpret the 1st question and think it is asking for the first time all 3 lights are turned off *simultaneously* and find the answer to be 60 sec.

Guess it is more of a reading comprehension problem than a math problem.

[robertjackson]

For all of those who are confused why this is "baffling" people, there is a simple explanation: the question can be misread in a few ways. For example, I started off thinking the answer to the first part was 60 seconds, because I thought it wanted the first time they all turned off at the same time. The question actually asks for the first time they simply are off at the same time, however. I assume that Rick's mistake is a similar one, but in reverse. He is reading the second problem as if it was asking for the first time all three light houses are off; however, the question asks for the first time that the three turn off at the same time. These two interpretations are very similar, and thus easy to confuse, which is why this problem is giving people a hard time.

[randallbosk-bearflagg]

This was a very easy problem and does not seem unusual for primary school.

From the comments however, it seems that some people have trouble understanding the question. I did not read the articles, so perhaps the wording of the questions were somehow ambiguous.

Key point of Question 1: All lights are off at the same time, regardless of when they turned off.

Key point of Question 2: All lights TURN ON at the same time, or in other words, all lights ended their off cycle at this time.

This is not the same question repeated!

Some answers appear to be answering the same question twice, once for when they are all off together (from 5 to 6 seconds) and then for them all on together (from 0 to 3 seconds technically, but assuming the next time, it happens again from 24 to 25 seconds) .

Other answers appear to be answering the question for when do they all TURN OFF together at the same time, which never happens.

Proof:

Given the time light C shuts off is:

C_Off = 5 + 10 x Nc, Nc = 1, 2, 3 ...

TIme for A and B going off are:

A_Off = 3 + 6 x Na, Nc = 1, 2, 3 ...
B_Off = 4 + 8 x Nb, Nb = 1, 2, 3 ...

There are no integer solutions for the conditional equation:

C_Off == A_Off == B_Off

Because C_Off must always be odd and B_Off must always be even.

Cheers!

[pokemaniacqolem]

I got the 2nd problem rather easily, but first one got me XD Misunderstood it as "exact moment every lighthouse turns off" instead :x

[nyimadrayang]

I quite like the question: "When is the next time they are all on" (after the first period of 3 seconds). In which case the answer is 24 seconds. At 24 seconds A and B will have just turned on (24 is a common multiple of 6 and 8)... but C will still be on (it turned on at 20 seconds and will turn off again at 25 seconds).

Lots of extensions possible, for instance:

The first time they are all off for 3 seconds in a row is at 45 seconds.

The next time they are all on for three seconds in a row is at 72 seconds.

If you map out all the seconds from 0 to 120 (on a spreadsheet or colouring in squared paper etc.) you get an interesting pattern which is symmetrical about 60 seconds (but inverted).

[vaprin]

This problem is much more intuitive than most problems you have featured so I would say that it's pretty easy.

[tomdekler]

These puzzles make me feel I somehow missed a few math lessons as a kid, lol.
I don't recall ever being told how to calculate the lowest common denominator. In fact I still kinda struggle with it, and I have no idea how you'd explain that formula to kids of that age.

I had the same realization when I noticed y'all factoring a polynomial as if it was the most instinctive and easy task in the world.

[mastergx]

Got the first part straight away. Made a mistake with the second part forgetting about the full cycle - I thought it was the least common multiple of 3, 4 & 5 (60).

[thes]

I just put it into Desmos
A = {mod(x, 6) < 3 : 3}
B = {mod(x, 8) < 4 : 2}
C = {mod(x, 10) < 5 : 1}

[BettyBoop]

My answer is 6 seconds for the first, and 25 seconds for the second part. I utilized an Excel spreadsheet, putting in 1's in rows that relate to the "on" times for each of the three lighthouses; then I continued putting in the appropriate number in three rows. For the 3-second light, I put in three 1's, then three 0's . For the 4-second light, I put in four 1's, then four 0's in the next row; then for the 5-second light, I put in five 1's, then five 0's, in the row under the 4-second row. I continued the numbers across the spreadsheet -- so I had three rows. When the columns showed only 0's for each light (Column 6), then the lights were off. When the 1's in all three rows were all in a column, then the lights were on. The "on" time was in the 25 column, so in my opinion, all three lights would be on at the 25-second mark.

[springboard]

I didn't have time for the math but I made a graph (similar to yours) and all the lights are on during the 26th second. The cycle for all three lighthouses repeats after the 120th second.

[ilsontfouscesromains]

When presenting a problem that had "baffled" parents (like this one but not only on this channel!) it'll be nice to present a suggestion of why it poses problem. For example as I see it, parents maybe were searching for the moment the three lighthouses turn off at the same moment (and never find it)

[thassalantekreskel]

I have to admit, the first part got me. I had thought it was asking when the lighthouses would all turn off simultaneously, not when they would all happen to be unlit at the same time. That said, here's how I figured it:
I began with the second part first, and doing this in my head, I kept it as simple as possible. Like in the video, I realized each cycle consisted of the full duration of the light being on and off added together. Then I asked when the other two cycles would align with an even 10 for the longest light, which is 30 and 40 seconds. Then it's simply figuring when those two larger cycles would align, which is at 120 seconds. This I mentally labeled a metacycle, and it occurred to me that in all likelihood the alignment of darkness would occur halfway through each metacycle, having the same period between all lights turning off at the same time as the cycle for them all turning on, placing that event at 60 seconds. But typing that out, I see that can't be the case because that's an even 10 seconds, which is when the third light turns on. So looking a little deeper, the third cycle's dark phase can be looked at as any number of seconds for a light cycle plus five. In other words, it will always be an odd number, and therefore impossible to align with an 8-second cycle, which will always be even. Had I seen that before watching the video in its entirety, I would likely have realized the trick in the first question.

[chinareds]

I just made a table. Probably what they expect primary school students to do. If we say the moment all 3 came on together is t=0 seconds, then they are all off at 5<t<6. The cycle takes a full 120s to repeat, so at t=120 they all come on together again.