100 Doors Puzzle || Hard Puzzle for Genius minds

There are 100 doors in a row. They all are initially closed.
A person walks through all the doors 100 times and every time he toggles the state of speific doors. that means he changes
the state of the door,. if the door is closed, he will open it.. and if its opened then he will close it.
In each walk he selects specific doors to toggle, in the following manner.
In his first walk, he toggles all the doors in multiple of 1.. that means he toggles all the 100 doors.
In the second walk, he toggles the doors in multiple of 2.
In the third walk, he toggles the doors in multiple of 3. that means, he will change the state of door no.3, 6, 9, 12... and
so on.
In the fourth walk he toggles doors in multiple of 4.
This way he walks for 100 times.
The puzzle is : What door numbers will remain opened after 100 walks.
(Remeber that initially all doors are in closed state)
This video explains this puzzle the best way. You will certainly learn the optimal way and the best approach of solving difficult puzzles.
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You can follow my facebook page of Logically Yours( Mohammed Ammar ):
/ mohammmedammar

[@lokeshreddy212]

Very difficult puzzle, but your explanation is very detailed. Thanks a lot for making this video....

[@alandouglas2789]

Mangoloads all the squares...

[@mohammedsiddiqui3062]

Yes this was a great explanation, and very easy to follow. How interesting it is that something can be so complicated and so simple at the same time.

[@medipallinarendrakumar1351]

Well ఎక్షపళైణైడ్

[@kiranvootori8101]

I can imagine how difficult making the animation would have been and appreciate how it makes understand the problem and approach to the solution so easy. But, I loved the door opening progress bar more than anything else - very creative.

[@bhaveshkhanted7087]

After completing the puzzle by writting all the factors till 100. Me : i am the most clever in the world. After video explanation.. Me : Oh , just missed that.

[@kanishkkrishna9983]

Really amazing explanation.Actually I was asked to solve this using programming.I even solved it in C++ by factor of a no. You are a genius sir. I never thought about square logic.

[@LOGICALLYYOURS]

Thanks Kanishk :)

[@jaygreen7494]

I love this type of puzzle, because you feel clever if you figure it out, but if you have to see the explanation it seems so simple and you wish you'd thought of it.

[@k.g6907]

ya this is always the case with me

[@shauryaagarwal4556]

This came in my Olympiad yesterday and today this is in my recommendation!!!! Aliens Confirmed

[@shauryaagarwal4556]

@Adam Romanov It was not specifically an Olympiad It was sort of a competition for a tour of IIT Kanpur conducted by students studying in IIT Kanpur

[@shauryaagarwal4556]

@Adam Romanov Thanks bro

[@udayadityabhattacharyya7496]

Though it is a known puzzle with 1000 doors, your explanation is superb.

[@qusuto]

Fortunately I was able to find this solution in today's codeforces contest.

[@manpowervicky]

I solved the puzzle using MS Excel, The doors open are - 1,4,9,16,25,36,49,64,81,100. Totally 10 doors

[@TigruArdavi]

Im too lazy, that's why I so love spreadsheets. Quickly drew up an excel, and instantaneously realized it's only square numbers remaining. That's the solution, so I didn't inquire it any further, appreciate your approach and explanation.

[@girlnado]

Kinda proud of myself on this one. I usually suck at these problems but I was able to work through it on my own, find the patterns, and get the right result. Very fun!

[@gopinathnatarajan6895]

It's really an awesome one. When you explained perfect square thing with this extraordinary animation. I am just wow! Thanks for making our brain smarter

[@nidadursunoglu6663]

I did it and found "all of the numbers which have even powers of their primes will be opened". Than i watched it and realized this defines the perfect squares 😅

[@LOGICALLYYOURS]

Glad you solved it on your own 😊

[@yasharya9765]

I solved it in 15 min

[@jaynilsheth1668]

If there would have been 'n' doors then in nth round number of open doors will be given by: greatest integer of √n that is : [√n] . You can check this by putting different values of n. You have solved for n=100.

[@gurshaan4848]

This puzzle was difficult, but after seeing this video I am able to analyze it,thanks Ammar

[@dwstyagi]

this puzzle is in the CtCI Book... but their explanation wasn't that good... thank you so much for explaining that briefly... Keep on doing that... Don't Stop

[@joneikholm]

public class DoorsToogled { public static void main(String[] args) { int size = 100; boolean[] doors = new boolean[size + 1]; for (int multiple = 1; multiple

[@SourabhBhat]

Using a computer program to simulate the puzzle: ----------------------------------------------------------------------------- numDoors = 100 doors = ['✘'] * numDoors for stride in range(1, 100+1): for i in range(numDoors): if (i + 1) % stride != 0: continue if doors[i] == '✘': doors[i] = '✔' else: doors[i] = '✘' print(doors) ----------------------------------------------------------------------------- Output: ['✔', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✔', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✘', '✔']

[@greninjamariokartpokemonfan]

The doors that will remain open are Doors 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. How do I know this? Well, opening or closing a door a certain number of times determines that door's factors. All doors that remain closed have an even number of factors, while those that remain open have an odd number of factors. Lo and behold, perfect squares are the only numbers with an odd number of factors. Here are how many factors each of the ten squares has, and what each factor is. 1: 1 (one factor) 4: 1, 2, 4 (three factors) 9: 1, 3, 9 (three factors) 16: 1, 2, 4, 8, 16 (five factors) 25: 1, 5, 25 (three factors) 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine factors) 49: 1, 7, 49 (three factors) 64: 1, 2, 4, 8, 16, 32, 64 (seven factors) 81: 1, 3, 9, 27, 81 (five factors) 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 (nine factors)

[@abhishekjadhav2363]

I solved it by reducing it to 10 doors and doing the calculation and found the pattern that 1 4 and 9 are open. guessed it correct that squares will be opened in the end.

[@Name-js5uq]

Wow, you did a really good job explaining it and it was easy to understand.it was well worth the time to do it right and explained it so very carefully I fully understood it. Excellent excellent job you did ammsr on this video.

[@LOGICALLYYOURS]

Thanks Steve for the appreciation... It really boosted my passion to make more videos. :)

[@subodhkhamkar2930]

Good visual explaination. I found the solution by programming and surprised to find that it's the square. Which highlights the importance of correct thinking over computer program

[@fascination1461]

Superb you are logically genius

[@lokeshkoliparthi9268]

At first I didn't get the logic of perfect square but at 3:40 I got the logic. Thanks for the puzzle 🙏

[@scyllasemusic2254]

I thought I would need paper, but, I used the picture in the video to visualise it easily instead. Then it came to my mind that all except 1 were closed as I thought others had even factors. But, I realised that one of the factors of a square number (the root) is multiplied by itself instead, and the person will only toggle the door once. So, instantly figured it out, the opened doors are square numbers Solved in 1 minute, well, not too hard, and also requires luck

[@Vicky-dp1fy]

Awesome........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... Bro!

[@Vicky-dp1fy]

Mind blowing

[@viktorsmets29]

I really liked this puzzle. It is comparable a locker riddle on TED.

[@nikhilramteke142]

Excellent Work!!! And very well explained. Thank You.

[@MrRobertpaul80]

Loved the explanation. Very very well done video! Thank you!

[@IroxGamer]

I saw many explanations but i could not understand why only perfect squares,,, but with this video i was able to understand THANKS A LOT

[@LOGICALLYYOURS]

Glad to know that my explanation helped you...

[@priyanshiagarwal2291]

I solved the puzzle and that too by same logic... practising...permutation combination really increases iq😄😁

[@gharud_xd6812]

Today interviewer ask me this puzzle and my mind said what's up logical people's 😂

[@ramaclassy8394]

thankyou sir very good explanation ...

[@ammaryasser2726]

i love his name first time to run into my name on youtube

[@LOGICALLYYOURS]

By the way... my son's name is Yasir :)

[@vickyca9313]

Very nice puzzle. I solved this using MS Excel. It was surprising that I cracked it in 10 mins by framing a formula in MS Excel

[@LOGICALLYYOURS]

Thanks Vicky.... good to know that you made use of excel formula.

[@amol1539]

Check for ten gates and you get 1 4 and 9 open... And you get the series logic!!

[@MUJAHID96414]

This isn't difficult 1st door will open and prime number door closed.

[@pappalaabhiram1670]

Superb Are you solving this problems on your own or with any reference .

[@keerthanasharavanan8169]

Thanks a lot for your video and explanation☺☺☺

[@ridarehman4727]

Got it Ammar By Rida 😊

[@vaibhavbaldeva9076]

It was fantastic

[@FurryCombatWombat]

My initial thought before watching rest of video: Only perfect squares will remain open because all other numbers have an even number of factors

[@FurryCombatWombat]

heck I was right now people are gonna think I cheated ;-;

[@cyburntormund6310]

A quistion ask me by an interviewer in a company that ' a Bear is at top of a hill (hight 100 meter) and fall down from hill in 10 second what is colour of BEAR .please answer this question sir

[@LOGICALLYYOURS]

Color should be White.. it's a polar bear(white in color).. it's found in Arctic Circle(north Pole of earth).. at poles the gravity is 10m/s. So it falls in 100/10 = 10 seconds (t = distance/speed).

[@faridfares8338]

Great but my english is so poor that l did not understand all the explanation

[@theletsplayprincess3894]

Those can't be the only open doors, can they? Feels like their should be more....

[@justingolden21]

Commenting before I see the explanation: Since all doors start closed, the open ones will be toggled an odd number of times. Primes (and composites as well) are divisible by themselves and 1, which in cases except for 1, cancel out. For example, 5 is prime and only divisible by 5 and 1, so that's 2 numbers, which is even, so they offset. For composites, they are also divisible by themselves and 1, however they can also be divisible by other numbers. 12 for example, is divisible by 2,3,4, and 6 (and 1 and 12). Since that's an even number of divisors, 12 will also stay closed. Since multiplying numbers pairs two unique integers together, unless the multiplied numbers are the same, the only numbers which have an odd number of divisors (and will therefore be left open) should be squares, such as 9, because 3 and 3 are the same number, so 9 has an odd number of divisors and will be toggled an odd number of times, and be left open. 1 is also a square, so this covers that case as well. 18 is divisible by 2,3,6,9, and even though it's divisible by 9 (a square, 3*3), it's divisible by an even number of numbers, as is 50 for example, where 25 pairs with 2, and 5 pairs with 10. 27 is a cube, and is divisible by 3 and 9, which is also an even number. Final answer: perfect squares only.

[@arshdeepsingh1348]

I solved this by starting with smaller set till 16, it makes an Arithmetic progression, 1open 2close 1open 4close 1open 6close ie after 1 open the close count increments by 2 So now if we do the sum of AP + open doors so that it is equal to 100 the n comes out to be 10 which is the answer

[@madhavaellamsetty1104]

I did not understand

[@arshdeepsingh1348]

@@madhavaellamsetty1104 Assume there are 16 doors and now you do the open close thing starting from 1st door then multiple of 2 then 3 so on till 16th door After doing this open close thing at the end you will see that 1st door is open 2nd and 3rd door will be closed, 4th door is open, 5,6,7,8 th doors are closed. 9th is open, 10,11,12,13,14,15th doors are closed and 16th is open. So it makes and AP 2,4,6,8 of consucutive closed doors and 1 open door in between them. You have to add this AP and the open door between them till the sum is 100. 1+2+1+4+1+6+1+8+1+10+1+12+1+14+1+16+1+18+1 = 100 So the 1's are the open door and the count is 10

[@madhavaellamsetty1104]

@@arshdeepsingh1348 wt does " AP" mean

[@JillC]

@@madhavaellamsetty1104 Arithmetic progression

[@BALAKRISHNAN-wb9hw]

Very hard puzzle. But really good explanation sir

[@samtabatabaeian2793]

a very good mathematical question I found out the answer in a very long time,but it really challenged me Thanks for your beneficial video

[@mattethebest1]

Squares?they have odd number of factors...

[@sameerkashyap4620]

Was asked this in an interview today, flunked grandly. Thanks tho, now I know how this works

[@BRIJ1501]

which company bhai?

[@sameerkashyap4620]

@@BRIJ1501 Juspay

[@BRIJ1501]

@@sameerkashyap4620 are you in IT domain..

[@onkarpitti767]

Actually it took me 1 hour for getting this As i took first approach writting all the factors And then i saw you video 😂 But really quite awesome 👌👏

[@anujkothari2134]

Awesome puzzle....same will be true for if we keep 10 doors and 10 times to toggle??right

[@captainjacksparrow690]

me ( when i saw the starting of video.) : The doors are all closed , then the puzzle is solved.🕷 me(after completing video) : swallowing saliva In fear.👻

[@baggiojoset]

Did it in 2 minutes I Checked it for only 10 doors I found out 1,4,9 are open at the end. It is because they are perfect squares.

[@suprajareddy2964]

super explanation what a logic 👌👌 really you are great thanks for making this video sir

[@LOGICALLYYOURS]

So nice of you

[@malahalder8534]

Ammar you are amazing bro. I am late to check out your channel but then also your puzzles really teaches to be rational and logical..... Love you

[@LOGICALLYYOURS]

That's all I needed.... a little appreciation boosts my passion to make more and more videos.... get ready for this weekend puzzle :) Many thanks Mala

[@malahalder8534]

Yes...

[@awfulprogrammer619]

In first ten number 1,4,9 will be opened(as 4 got closed by 2 and opened by itself, same as 9 got closed by 3 and opened by itself). So, we have Arithmetic sequence as.. 1,4,9,16,25,36,64,81,100...

[@shaheedsukhdevnagar24shaku98]

I also figure out this solution by this method. 🙂🙂

[@shaheedsukhdevnagar24shaku98]

Your trick is amazing 👍🏼👍🏼. But I think u missed 49.

[@Ressuu]

Spent 15min on the puzzle. It was clear there had to be a rule and dealing with the first few numbers it quickly becomes apparent that odd number of factors is the key. How can some people manually calculate all the odd factors up to a 100 and not see this is amazing XD

[@yairkaz]

Most people create code to check this, I personally did and I know many else did as well

[@d4django]

Very interesting. Excellent explanation. And an interesting fact about squares.

[@XJWill1]

Your explanation would be clearer if you used the word "divisor" instead of "factor". When someone says "factor" the first thing many people think of is a factorization. For example, 36 has the prime factorization 36 = 2^2 * 3^2 but what we need to know for this problem are what are the divisors of 36, i.e., what numbers can 36 be divided by with 0 remainder? Those are: 1,2,3,4,6,9,12,18,36

[@gq7435]

Awesome logic. But I think I will just write code to find correct doors than do all this analysis.. that would be a lot faster 😀

[@Nuclearburrit0]

Programming is fun

[@rakshit_jain_sahab]

The excellent explanation. You are a genius

[@deeptanshusurya6399]

It takes few minutes for us to watch the video but for sir it takes a lot of hard work and patience to create such an amazing , informative and brain storming videos. A lot of appreciation to you sir.

[@sushantagawane8683]

Very good explanation. Thank You !!!

[@navdeepsingh6204]

Incredible approach..... Totally out of the box ..... Great to watch such a solution....keep it up

[@padmavathynarayana7089]

Sir wat about the prime numbers sir Will not the doors of prime numbers be opened??

[@LOGICALLYYOURS]

Prime door numbers would be closed because they would be visited twice.. example: Door 7 would be visited on walk1 and walk7.

[@sakeenakalam5373]

You are genious

[@Agnihotri798]

I did it by own

[@priyadharshinikumar4007]

Such awesome explanation....tq bro

[@kk-fj9mi]

It will be all perfect square less than 100

[@vembukarthickt8809]

#TED RIDDLES *************************************************** sarcacism

[@DarkZero757]

I solved it using c++

[@dashl5069]

this is a fun logic puzzle but really not that difficult, my class was given this problem in 4th grade and almost everyone was able to solve it in 30 mins or so.

[@manusarda]

I have viewed this puzzle 2 times before on other channels but never understood the concept. Although I mugged up the answer that perfect squares will remain open. But now after seeing your explanation there is conceptual clarity. Great explanation 👍👍

[@abhinandan3175]

My teacher just taught me this question a few days back....😉😉 And the answer is 10!!

[@scyllasemusic2254]

Yeah, that's the amount

[@brianmagicman3556]

That's obvious, seeing as the answer on here was 10 anyway

[@sangeethamaryr368]

Nicely explained sir Thank you

[@icebro5059]

I like how this video uses the SpongeBob font.

[@suryarb4843]

Single question gives.teaches a lot of information and hints.

[@InfoPanel7902]

Explained beautifully

[@sriramyatoleti]

Really appreciate the presentation. Must have taken a lot of effort, but beautifully illustrated!

[@RadheRadhe-js8co]

Sir cleared it Without seeing the solution

[@aayushjalan712]

I actually wrote the code to count the factors... anyway, nice question and solution Excellent

[@LOGICALLYYOURS]

Hi Aayush... can you please share the code.. just need to check if you wrote it based on algorithm or any other logic.

[@crickster1333]

Sir what about 12, it having odd number of factors.

[@arshdeepsingh1348]

1,2,3,4,6,12 which is 6

[@brwler1]

subscribed

[@komedy3001]

A very good puzzle and explanation tooo

[@ameersuheal8070]

Is it true??? I still din get it

[@williamjoannes7197]

nice puzzle, the only thing I could think of was that all primes are closed in the end (but actually many others all also closed)

[@panchalibaishya749]

Awesome explanation 😍

[@olerask2457]

I solved the problem using prime factor decomposition. Any natural number N may be written uniquely as p1^n1 * p2^n2 * ... * pk^nk, and the number of divisors of N is then (n1+1) * (n1+1) * ... * (n1+1), which is only an odd number if ALL n1, n2, ..., nk are even. But then each factor p2^n2 is a perfect square, and as any product of perfect squares is also a perfect square, the condition for an open door is that N should be a perfect square. The approach in the video using matching of divisors is simpler than mine, and also very intuitive :-).

[@praneethkuraparthi3165]

However i came with the same approach till the number of factors to be odd.There is an other explanation to this. we know that the number of factors for a number expressed in the form of prime numbers such as 2^(a1)*3^(a2)*5^(a3)..*x^(an). is (a1+1)*(a2+1)*(a3+1).....*(an+1)..let it be equation(1).Where 'x' is the largest prime number. And we know that odd*even is even, even *even is even, only odd*odd will be odd number, so every term in equation(1) is odd number i.e, (a1+1),(a2+1),(a3+1)...,(an+1). are all odd numbers so we can say a1,a2,a3,a4,....,an are even numbers(since if a1+1=odd, a1=odd-1 would be even) in case of odd number of factors. so this number should be a perfect square because it can be expressed in a form of (2^(a1/2)*3^(a2/2)*5^(a3/2)..*x^(an/2))^2. a1/2, a2/2,a3/2,....,an/2 are integers (since a1,a2,a3,....,an are even numbers ) we will get a integer square. so we can conclude that doors opened after 100 repetitions would only be perfect square numbers as mentioned.

[@gharud_xd6812]

Thanks bro this is made my interview ❤️ Interviewer ask same Puzzle 😂

[@sankhya_0461]

haven't looked at the video but the answer has something to do with a number of factor as 2^p.3^q.5^r... is the prime factorization of a number then it has (p+1)(q+1)(r+1) factors. But this will yield the factors as even and even number of factors means the door will be closed. The only way that they remain open is if they are squares as they are the only numbers with odd no of divisors. JEE aspirants should've done this during P&C and this question should be a piece of cake Yes I was right.

[@hocaponttas]

It took me like two hours and I still solved it partly wrong... 😂 My answer was 1, 4, 9, 12, 25, 30, 60, 64, 80, 81, 84, 90 and 100. I'm not sure what happened

[@apnapython]

very nice explanation easily understand the concept behind this puzzle.

[@LOGICALLYYOURS]

Thanks Aman :)

[@christianromero1334]

Made a table of door vs walk. Used 1s and 0s to signify open and closed. There was a diagonal pattern 100 10000 1000000. The difference between Every door number with the last zero went up by 2 every time. Door 99 was one of them so 100 must have been 1 (open). His way was better 🙄

[@manansheth3084]

Perfect squares will be open because they have odd number of factors.

[@JEManishkumar]

i just alnayzed this question with 10 door , then i analyze it for 20 , i got 1,4,16 are opened , i immediately said all perfect square no. door will remain open, analyzing with small no. makes any question easy

[@bharathraj435]

Just an another way to solve riddle having big numbers is first trying with sample. For example in this riddle assume 10 doors and 10 walks and see how many doors are opened .we'll get 1 4 & 9..from these numbers we can get idea that perfect squares are the answers

[@soumyanarayanbehuria6694]

Perfect square no.s only have odd no of factors..that's it's answer..nice question... thanks a lot..