Impossible Logic Puzzle from Indonesia!

[@XkcdEsperite]

This can be solved even more simply: If someone knows whether the numbers are being added or multiplied then they know the solution. If your number is greater than half the total (1010) then you know for sure that the numbers are being added together, and would know the solution. Alzim does not know the solution, so his number must be 1010 or less. Era does not know the solution, so his number must also be 1010 or less. Because he still does not know the answer there must be a way add Alzim and Era's numbers together to reach 2020, and neither one is greater than 1010, this is only possible when Era's number is exactly 1010.

[@Furkan-yv5ew]

There are only 3 conclusions you found, and (4,550) satisfies all 3 of them. And btw there isn't only one solution.

[@Starwort]

​@@Furkan-yv5ew4,550 doesn't work because the second person can conclude that it must be a product - they know that both numbers are ≤ 1010 and thus must be multiplied to get the answer

[@majordude83]

Yeah, it's much easier if you start with "x and y must be less than or equal to 1010, or else one of them would know the answer immediately".

[@YaacovIland]

That's a really nice way to solve it!

[@Furkan-yv5ew]

@@StarwortI know it can't be a solution. However, he claims that the only solution is when both numbers are 1010 based on the conclusions he found, but that's incorrect. That's what I'm saying. And you are right. The second person would be able to answer the question because he knows the first person must have a number that is a divisor of 2020, so the first person couldn't determine the solution initially. From there, the second person can identify the number the first person has when it is asked. I used this example to show that he hasn't figured everything out. He needs more conclusions. Prove me wrong.

[@NathanSimonGottemer]

Three logicians walk into a bar. The barkeep asks if they all want beer. The first says, “I don’t know.”The second says, “Me neither.” The third says, “yes.”

[@vt2788]

The third says without sugar

[@NathanSimonGottemer]

@@vt2788 actually it continues, - the barkeep brings three beers. A half hour later the barkeep asks if any of them want more. The first says, “I don’t know.” The second says, “me neither.” The third says, “no,” and the three pay the tab and leave.

[@vt2788]

@@NathanSimonGottemer what a twist

[@cynicviper]

​@NathanSimonGottemer I caught the "all" in the first part immediately, but had to read the second part twice to notice the "any". Great joke!

[@NathanSimonGottemer]

@@cynicviper ehe thanks. I do love a nerdy joke.

[@WilliamWizer-x3m]

you can even generalize it. let N being the number given by the host. if the answers given by the players are NO, NO, YES then the second player's number will always be N/2. why? if Y>N/2 then X=N-Y. (as Y>N/2 is the same as 2Y>N and that prevents multiplication) if YN/2 so first player would had known Y=N-X (as X>N/2 is the same as 2X>N and that prevents multiplication) in short. first player must be either N/2 or 2 and second player is always N/2.

[@JohnDoe-ti2np]

Right. In fact, if the first two answers are NO, NO, then the third answer must be YES. So we can modify the puzzle: We don't tell the reader that the number on the card is 2020; we just say that the number is publicly shown to both Alzim and Era. We also don't tell the reader what Alzim's third answer is; we just ask, "What did Alzim say when asked if he now knows Era's number?"

[@skya6863]

​​​@@JohnDoe-ti2np i don't think that holds true if the number is not divisible by 2

[@JohnDoe-ti2np]

@@skya6863 If N is not divisible by 2, then the first two answers won't be NO, NO. If they were, then that would mean that both Alzim's number and Era's number were factors of N. So Era, upon learning that Alzim's number is a factor of N, would know that Alzim's number has to be N divided by Era's number. This is a contradiction.

[@skya6863]

@@JohnDoe-ti2np yea that's true! Cool stuff

[@foogod4237]

@@JohnDoe-ti2np Or you could turn it into a _real_ trick question and just ask "Does Alzim know what Era's number is?"

[@deerh2o]

When we know that Alzim's number is a factor of 2020 at the beginning, we also rule out 2020 as the sum would make Era's number 0, which is impossible.

[@mxmdabeast6047]

We rule out 2020 in the second step as it cant be used as a sum AND product. But, in the first step, its a valid number. This is because 2020 * 1 is a valid product. We only know that the number is a factor, not if it is a sum or a product.

[@aaronbredon2948]

​@@mxmdabeast6047but if alzim's number is 2020, alzim then knows that Era's number must be 1, so you don't get to the second step.

[@mxmdabeast6047]

@@aaronbredon2948 Ah, gotcha. I was thinking about Era's number, not Alzim's. Makes sense

[@Rubbinghandsschemingsomething]

Era should've been able to guess it

[@phoenixarian8513]

Well 2020 is still in the possibly list. If alzim has a 1 and the card says 2020 he can assume era's number to be 2020 (product) or 2019 (sum). However if alzim has 2020 and the card says 2020 he would immediately declare era's number to be 1 since 0 is not valid as the precondition "greater than 0".

[@bg6b7bft]

1) if Alzim's number isn't a divisor into 2020, then he would know it was a sum, and would know the answer. Therefore his number must be 1, 2, 4, 5, 10, 20, 101, 202, 404, 505,, 1010, or 2020 2) If Era's number isn't a divisor into 2020, then he would know it was a sum, and would know the answer. Therefore his number must be 1, 2, 4, 5, 10, 20, 101, 202, 404, 505,, 1010, or 2020 If his number is 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, or 2020 he would know it was a product, and would know the answer. However, if is number is 1010 it could still be either a sum or product. The answer could be 1010 or 2. 3) Since Era didn't know, Alzim now know's Era's number is 1010.

[@mohitrawat5225]

There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he would have been confused.

[@dattaprasadgodbole]

My logic was the same. ]

[@dimitriskontoleon6787]

But still why said yes era on second round? Is still between 2 and 1010

[@natchu96]

​@@dimitriskontoleon6787If Era had 2, then Alzim can only have 1010 or 2018; however, Alzim is going first. Only if Era had 1010 can both players be uncertain on turn one. Since Alzim obviously knows his own mumber, if he had 2018 then he would've known Era had 2 on the first turn, as you can't multiply 2018 by a whole number and get 2020.

[@ejrupp9555]

@@natchu96 I think DIMITRI is looking from Era's perspective going 2nd, as the 2nd round, and also having the answer. I translate him as saying How can Era say yes on the second turn, (which is Era's first.)

[@violetfactorial6806]

This one felt good to solve. There's a lot of information given by the fact that they can't immediately tell that it's a sum. The question is more like "can you tell if it was a sum or a product?" Since the answer was 'no', it was obvious that their numbers must have the potential to both sum to 2020 AND to multiply to 2020. After some thinking, the only pair of factors of 2020 that could ever sum to 2020 are 1010 + 1010, so that must be the answer. It follows that Alzim's number is either 1010 or 2.

[@ARex545]

I did it in a pretty similar fashion but arrived at the solution pretty quickly. Here's my trick-> We need to ask ourselves what will create ambiguous situation even if we had all the information. An ambiguous situation only arises when 2020 can be formed using both the multiplication and summation of two numbers. This allows us to come down to factors of 2020. Now again going by the same logic as earlier, what can still cause us to be ambiguous even though we have the information? We can clearly see 2×1010 and 1010+1010 are the only numbers that would still create confusion while solving the puzzle for Alzim and Era. The process mathematically is the same, its just that logically I find it easier in these types of question to create a forceful ambiguous situation even though we have all the information. Nothing groundbreaking just helps me solve more quickly!

[@nuclearmedicineman6270]

That's what I did. Both numbers have to be factors of 2020, and both summation and multiplication have to be available or the puzzle never even gets to the third question. There has to be confusion on the second player's turn or the game's already over. Any non factor number, and the game's over on the first question. Any factor that doesn't offer confusion on the second player's turn and the game is over. The only logical solution is 2x1010 or 1010+1010. Only number that actually goes as far as the third question.

[@mohitrawat5225]

​@@nuclearmedicineman6270Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@DaveBermanKeys]

Let me just randomly pick some numbers to show you why they don't work. I don't know...how about 420 and 69?

[@ZelenoJabko]

How about no?

[@tirlas]

meme math

[@joeschmo622]

That's exciting...

[@muskyoxes]

You couldn't say 42, because that's the answer

[@arc-sd8sk]

nice

[@Indian_Ravioli]

1010..... This took me more time than i initially gave it credit for, because I went from thinking it was easy to discovering why it's not that easy, to finally finding the actual answer😮‍💨😌. I'm glad I follow your page.👏👏

[@cts3md]

I think you can definitively say that Eras number is also 1010 and here’s why I say that: When Alzim is first asked what Era’s number is, he only knows that his own number is 1010 and that 2020 is either the sum or product. From that he can say that Era’s number is either 2 or 1010 but not which, so he says “no.” Next Era is asked. If his number were 2, then he would realize that Alzim’s number could’ve been either 2018 or 1010, but if it had been 2018 then Alzim would have known that Era’s number had to be 2 and he would have answered “yes”. From that Era would know that Alzim’s number had to be 1010 and so Era would’ve said “yes.” If Era’s number was 1010, though, he would know that Alzim either had 1010 or 2 but not which and so also said “no”. Therefore the fact that Era didn’t know Alzim’s number tells Alzim that Era’s number had to be 1010.

[@nahommerk9493]

That's what Presh said! Era's number is definitely 1010 but Alzim's can be either 2 or 1010. I think you're confusing Alzim and Era

[@jkoh93]

@@nahommerk9493 Presh is wrong. Both numbers are 1010. Alzim's number cannot be 2

[@fernandodavidfelixfalcon2351]

​@@jkoh93Alzim can have a 2... 1. Alzim answers NO because he doesnt know if Era has a 1010 or 2018 2. Era (with a 1010) answers NO because he doesnt know if Alzim has 2 or 1010 (in both scenarios Alzhim would answer NO) 3. Alzim answers YES because now he knows that Era doesnt have the 2018 (otherwise he would have answered YES)

[@HeirToPendragon]

Quick solution for those that want it: Era's number must be a factor of 2020 such that Era's number * Another factor or 2020 = 2020 Era's number + Another factor of 2020 = 2020 This is why Era doesn't know Alzim's number. Once Era doesn't know, Alzim just has to find the only numbers that could work, and in this case there is just one: 1010 Because 1010 * 2 = 2020 And 1010 + 1010 = 2020

[@defnotisaiah3210]

love your videos presh but ai art is meh

[@MelomaniacEarth]

True😂

[@gmdFrame]

True😂

[@TheScienceGuy10]

True😂

[@francomiranda706]

Would you rather him get some stock images off of the internet? in either case, no artist is being paid and the images are serving their purpose.

[@defnotisaiah3210]

@@francomiranda706 i think ai art is ugly, so to answer your question, yes. you’re acting like he hasn’t made hundreds of videos beforehand, there would be no change

[@QuantumOverlord]

I haven't looked at the solution yet. Here is my solution: Alzim does not first respond by saying he knows. This rules out him having any number not in the set F (where F is any factor of 2020) because if Alzim's number is not in F then Era's number must be 2000-Alzim's number. So we know from the first 'no' that Alzim has picked a number in F. Era concludes the above so knows that Alzim has picked a number in F. Therefore we can conclude Era's number must also be in F (if the card is the product) or in F' which is defined as the set of 2020- each element in F. If Era's number is exclusively in F then Era would answer 'yes' since he could determine Alzim's number is 2020/Era's number. If Era's number is exclusively in F' then by similar reasoning he can answer 'yes' and determine Alzim's number is in 2020- Era's number. Given Era does not answer 'yes' we know that his number cannot be exclusively in F or F'. But since Alzim's number is in F it can't be in neither either. Therefore the only other option is that it must be in *both* F and F'. The only factor of 2020 that is in F' is 1010 since 1010*2=2020=1010+1010. Therefore Era's number is 1010 Alzim also concludes the above. However there is not enough information to determine whether Alzim has picked 2 or 1010. Alzim's number could be either of those options.

[@averagejuveenjoyer1994]

With person's number n other's number can be only 2020-n or 2020/n. If a person doesn't know, then both variants are possible, so his number is a divisor of 2020. Both dint know, so both numbers are divisors of 2020. The only way sum of 2 divisors of 2020 can be 2020 is 1010+1010 (since at least one of them must be not less than 2020/2: 1010 or 2020). This variant fits Let's consider that 2020 is a product. The second one knows that both numbers are divisors. If his number isn't 1010 then sum isn't possible -> the second knows both numbers at his turn - contradiction. Era's number is 1010

[@fjaps]

Much cleaner than the solution in the video!

[@mohitrawat5225]

Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@NeuroticNOOB]

​@@mohitrawat5225you didn't explain why Alzim's number should be 1010 at all.

[@dustinbachstein3729]

Nice puzzle! I found it rather easy, though. A few notes: 1. The same puzzle works for all even numbers instead of 2020. Listing up all factors of 2020 is also unnecessary; instead, it's enough to solve the equation 2020/a + 2020/b = 2020, where the 2020 cancels (that's why every even number works) and a=b=2 is the only solution. 2. The information that Alzim says "yes" in the end is irrelevant because he couldn't have given a different answer anyway. 3. Also, I don't think the extra condition that the chosen numbers must be positive is necessary. The puzzle should work the same way if any whole numbers were allowed.

[@matrenitski]

1. True 2. Totally! 😅 3. Natural numbers can't be negative by definition It is a nice puzzle though. At first looks hard, but then all clears up!

[@zer0mar322]

What I find the most interesting in this problem is the insight in hindsight: As soon as the 2020 card was shown, Era (having picked 1010) immediately knew that Alzim had to have either 2 or 1010 and Era learned absolutely nothing else about Alzim's number throughout all the rounds of questions! Alzim, having picked 2 or 1010 (we still don't know which), would have also known from the getgo that Era could have one of 2 numbers. If Alzim has 2, then Era has 1010 or 2018. If Alzim has 1010, then Era has 2 or 1010. Here is an alternate reality scenario with an ever so tiny change, that ends up in a very different result: What if mufti-saab questioned Era first instead? If Era was asked first if he knew what Alzim's number is, he would say "no", and if Alzim had picked 2, Alzim would immediately know that Era could not have 2018, otherwise Era would have immediately known what Alzim had, and conclude that Era has 1010. If Alzim picked 1010, Era not knowing what Alzim had would not have sufficiently narrowed it down for Alzim, and Alzim would say "no". Ironically, this difference in outcome would actually expose to Era what number Alzim had! Alzim would say "yes" if he had 2, or say "no" if he had 1010. So, if Alzim picked 2, Alzim's declaration that Era picked 1010 would tell Era that Alzim picked 2, thus both of them knowing each other's number in the end! And if Alzim picked 1010, Alzim's not knowing what Era has would give away that Alzim has 1010. Era then declaring that he knows what Alzim has would confirm to Alzim that Era also has 1010, and this is because if Era had 2, Alzim knows that ERA KNOWS that Alzim could not have had 2018 without knowing what Era has. Isn't it fascinating that asking fewer questions, but changing the order could have actually revealed MORE information to all parties? Alzim and Era strictly know more than outside observers like us do. Would there be a scenario in which they both know each other's number, but we don't know one or both? Hmmmmmm! I don't think so? Well, thank you so much for reading! I'm actually not 100% confident that my reasoning is actually logical, because, man, I didn't get enough sleep last night! If you somehow managed to parse through all the Alzims and Eras in that, I congratulate you. I'd love to see someone call me out on being wrong, because that showed they care enough about my rant.

[@GODWIN404]

My mind sleeps while watching 😂

[@Raven-Creations]

I really like this one. It requires being able to see the puzzle and the options from both players' perspective.

[@janjanssen5978]

I think the riddle works also if you don’t specify “greater than zero “ in the beginning.

[@TheRealScooterGuy]

No, it opens the possibility of there being two working solutions.

[@JohnRandomness105]

Once Alzim answers "No", Era knows that Alzim's number is a factor of 2020. If Era's number is greater than half of 2020, and not 2020, 2020 must be the sum -- he figures it out. If Era's number is less than half of 2020 or is 2020 itself, 2020 must be the product -- again, he figures it out. The only ambiguous situation is when Era's number is 1010 -- Alzim's number is either 2 or 1010. This reminds me of Mark Goodliffe solving a "Genuinely Approachable Sudoku". He kept breaking the puzzle on applying a rule leading to 5 + 7 = 12. The were other rules, and Mark finally broke through saying out loud, that we couldn't have 5 + 5 = 10 -- oh, but we could have 5 * 2 = 10! That was after several trials and crashes, and he got a well-earned dinosaur.

[@guilhermeteofilocachich4892]

I love Cracking The Cryptic! Two very great channels indeed.

[@JohnRandomness105]

@@guilhermeteofilocachich4892 Indeed!

[@philippecanepa4509]

The most difficult part for me was first to put myself at the place of Alzim and after at the place of Era taken into account the assumption of Azim and switch again to Alzim with the assumption of Era. No-No-Yes. The change of perspective is more difficult to perform than the mathematical logic.

[@Mateusz-Maciejewski]

Many of you realised that there are easier way to find the solution without writing down each of the divisors of 2020 and that it can be generalised to any even number N instead of 2020. So it can be a base for another riddle: Mufti says the number 20475. Alzim says 'No', Era says: 'Hey! Mufti, please use a calculator and start again.' By the way, the riddle can be shorthened to the implication: n>m>0, m | n, n-m | n implies n=2m.

[@johnschmidt1262]

I think you ended up doing it the long way. The key insight for me was if either person's number was over 1010, they would know that it has to be addition, as the product would be above 2020 (or * 1, another easy answer). But once Era knew that Azeem number was under 1011 he should have been able to divide 2020 by his number to get Azim's number (knowing the result was a product the only way two numbers under 1011 to make 2020), unless his number was exactly 1010, which is the answer!

[@WilliamWizer-x3m]

correct. that's the clue I used too. it can be generalized to use any even number. the second player's number is always half of that number so he can't know if the first player has his same number (addition) or a 2 (product)

[@mohitrawat5225]

​@@WilliamWizer-x3mAlzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@NeuroticNOOB]

​@@mohitrawat5225again you just copy pasted and did not explain why Alzim's number must be 1010. You just talk about Era.

[@amethonys2798]

​@NeuroticNOOB because no other option is possible. We can rule out literally every other option because both numbers HAVE to be less than 1011 and still be ambiguous on if it is product or sum from Era's perspective. Let's look at it from Era's perspective. Let's PRETEND he has 404 (and thus Alzim has 5). Both numbers are less than 1011, but since Alzim said "No" earlier that rules out sum as an option since if Alzim had 1616 he would've known Era's number. If Era had 404 the ONLY number Alzim could possibly have at this point is 5 and thus would answer "Yes". The ONLY way Era says "No" is if his number is exactly 1010 as Alzim could still have 2 or 1010.

[@NeuroticNOOB]

@amethonys2798 you're literally making a conclusion about era's number here like the other person that CLAIMED in the first sentence that we would know Alzim's number

[@asheep7797]

My working: Alzim: No. If Alzim’s number was not a factor of 2,020, he could quickly rule out the possibility of a product, an easy gateway to a “yes.” However, he cannot determine if 2020 was a result of multiplication or addition, and thus, Alzim’s number is a factor. Era, knowing this, has narrowed down the options to 1, 2, 4, 5, 10, 20, 101, 202, 404, 505 and 1,010. However, he seems to also have trouble determining the card, thus implying that his number is both a member of the afformentioned set, and the new set: 1,010, 1,515, 1,616, 1,818, 1,919, 2,000, 2,010, 2,015, 2,016, 2,018 and 2,019, meaning he picked 1,010. Although Alzim now knows what card was chosen (as he knows his own number), we can never know, as he (following the same logic as us), knows Era’s number was 1,010. Alzim’s number is either 2, or 1,010.

[@taflo1981]

After watching only the beginning of the video in order to be sure what "number" is supposed to mean in this context: This works with any even positive integer N (apart from 2) instead of 2020. By saying "No", you reveal that N is both a multiple of your number (vice versa, your number is a factor of N) and a sum of your number and a positive integer (meaning that your number is between 1 and N-1). So when the first person says "No", we know that their number X is a factor of N (but not N itself). Before the second person answers, they already have this information about X, so when they say "No", we can not only deduce that their number Y is a factor of N (but not N itself), but also that both N/Y and N-Y are factors of N, since those are - from the pov of the second person - the two possible values of X. We can ignore N/Y, since that is a factor of N as soon as Y is. But if Y were strictly smaller than N/2 (but still positive), then N-Y would be strictly larger than N/2 (but still smaller than N) and thus not a factor of N, iplying that the second person's answer would have been "Yes". So we know that Y=N/2. Since N is the sum or the product of X and Y, we have X=N/2 or X=2. Remark 1: When we look at N/Y for the second person, we immediately see that they couldn't have chosen Y=1, as that would imply that the other person's number was N, which we already excluded. This is why the argument fails for N=2. If you allow X,Y to be 0, then it also works for N=2. You just have to observe that if one of them chose 0, they would immediately have known that the other one chose N, as N is not zero and thus not a multiple of 0. Remark 2: "Number" could in fact just refer to "integer" (i.e. including negative values) and the argument would still work with a few slight changes. Indeed, if the first person answers "No", then X has to be any factor (positive or negative) of N. So if Y were a negative number, the second person would immediately know that N can't be X+Y, since otherwise |X|>N and the first person would have answered "Yes". Remark 3: If "number" were just to refer to rational or real numbers, the whole argument would break down and the only things we could deduce is that neither of them chose 0 as their number and that Y is not N (as otherwise the second person would have known that N = X*Y, since they would have already known that X is not 0).

[@guyhoghton399]

Almost right. It doesn't work when N = 4. Otherwise it does for any even integer > 4.

[@edsimnett]

Given A does not know immediately, his number must be one of the whole number factors of 2020 (otherwise it would be a sum situation- and he cannot have picked 2020 because he would know it was a product situation) So E knows that, so he knows A has one of 11 numbers. So given he says no as well, there must be ambiguity which can only happen if E has 1010, because A could have 2 or 1010, in every other E it would be obvious So A knows that, and knows his own number, so he knows E's number is 1010, but we cannot know what A has except that it is 2 or 1010.

[@mohitrawat5225]

Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@alanshand829]

This didn't take long to work out. Knew straight away that both had to have picked factors of 2020, its the only way that there could be any confusion between the sum and product. If Era had picked any other factor apart from 1010 he'd know 2020 was the product and been able to tell Alzim's number since there are no numbers big enough for it to be the sum. Since he can't then Alzim knows its 1010, meaning Alzim picked 2 if 2020 is the product or 1010 if its the sum.

[@michaelcarlton1484]

The first question is always going to be no since if they had non-factor numbers, they would immediately know it's addition and figure out the other number. From there, they simply wait for the second person to say yes or no since the second number COULD be a factor or not. If the other number is a non-factor then, again, they figure out it is addition and solve it. Once both people say no, they know it's multiplication and thus, solve it that way.

[@amethonys2798]

It is also only No for any number less than 1011. Anything over N/2 is instantly solved since it HAS to be the sum as well. This is also why Era's number has to be 1010 since if Era had like 5 or 404 he would instantly know Alzim's is the other since Alzim's number couldn't be 2005 or 1616 respectively.

[@MonsterERB]

I worked it out like this (prior to watching, of course): 2020 factors to 10 * 202 = 2 * 5 * 2 * 101 Prime factors are 2 2 5 10 Possible products: 1*2020; 2*1010; 4*505; 5*404; 10*202; 20*101 If ANY other number was chosen by Alzim, he would instantly know it was a sum and would answer yes Alzim does NOT answer yes, so his number is from the set (1 2 4 5 10 20 101 202 404 505 1010 2020) Era's turn; he also answers no so his number is also from the same set as Alzim, but it also must be part of a possible SUM (with both addends from the same set) since he answers no The only possible solution is that Era has chosen 1010; he doesn't know if Alzim has 2 (product) or 1010 (sum) Alzim's turn now; he knows Era's number is 1010. Alzim's number is either 2 (product) or 1010 (sum)

[@mohitrawat5225]

Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@guyhoghton399]

This problem could be set for any even card number _N_ greater than _4._ When both men learn that their opponent doesn't know whether _N_ is the sum or product of their numbers, each can tell that his opponents' number must be a factor of _N ≤ N/2._ But now the 1st man can deduce that his opponent's number cannot be _< N/2_ otherwise the latter would have realised that _N_ could not be the sum. Therefore he knows that the 2nd man's number must be _= N/2._ Choices of _N:_ _N_ must be even for _N/2_ to be an integer. _N_ cannot be _2_ because then there is no factor of _N_ that is _< N/2,_ which removes any doubt from the 2nd man. Also _N_ cannot be _4_ for a different reason. Can you see why?

[@FarrizWil]

It's been a long time since the last time I watched many of this channel content. Finally youtube recomend me Marty's bigger pizza and now this is the next one. Greeting from Indonesia.

[@guilhermeteofilocachich4892]

Fantastic puzzle. I was able to get it on my own, after a few minutes thinking. Will surely share with friends. Thanks for bringing it on the channel, Presh!

[@ktomeir]

I describe the logic as: If one chooses a number greater than 1010, he will immediately know that that 2020 is a sum, so Alzim's first "no" means x=1010.

[@AdrianColley]

This was a really good one. I couldn't figure it out until 8:14 into the video.

[@MiaVilleneuve]

i like how you picked 420 and 69 as example numbers

[@howareyou4400]

y | 2020 => ty = 2020 (2020 - y) | 2020 => (t-1)y | ty => (t-1)| t This is ONLY possible when t = 2 therefore y = 1010

[@realedna]

The solution comes rather from the fact, that THERE IS A solution to this formulation in addition to the case, where the revealed number would be the product to show, why Era is still unsure. So Alzim knows Era's number in either case, but Era does not know Alzim's and which case it is.

[@kandaffar2344]

Era made a mistake. To win the game, he had to say Yes (Lie) and make a 50/50 guess, because he should have known he would lose on Alzim's next turn.

[@DavZell]

Era would have known on his first turn. He would known Alzin would only have questioned whether Era had a 2 or 1010 if Alzim himself didn't have a 2. Therefore, Era would know Alzim didnt have a 2 and they both had 1010's.

[@frankzunderer7730]

"Alzin would only have questioned whether Era had a 2 or 1010 if Alzim himself didn't have a 2" - True, but in case Alzim had a 2 he would have questioned whether Era had a 1010 or a 2018, so his first answer would still have been "no".

[@Tronnixx]

I paused the video before watching the rest of the video. Correct me if I'm wrong. I think that based on the question stated by Mufti. I assume that 2020 is the product of two numbers, the card randomly revealed to us. The card that I think is randomly hidden is the sum of two numbers card. The question mentions there are two cards, one of them is hidden, and the other is shown. To answer the two questions: What is Era's number? Era's number is either {2,4,5,10,20,101,202,404,505,1010}. Other possible sums of Era's number are: 2 plus 1010 equals 1012. 4 plus 505 equals 509. 5 plus 404 equals 409. 10 plus 202 equals 212. 20 plus 101 equals 121 Based on this entire set of numbers, 1010 or 2 is most likely to be Era's number because you can add 1010 by 1010, which gives the sum of 2020 and you can multiply 1010 by 2 to get 2020. What can you conclude about Alzim's number? 2020 could be a sum or a factor of 1010 multiplied by 2. Alzim's number could be 2 or 1010.

[@mohitrawat5225]

Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@SeriesGamer2008]

but if we imagine 1010 as both of their number, it wouldn't make any sense. because in that case, alzim still wouldn't know era's number in the third step. the only way he can know era's number after the second step is if 2020 mines his number is not a factor of 2020 and therefore after finding out that era's number is a factor of 2020, he could rule it out and so 2020 is only the product of the two numbers and so, alzir's number can only be 2.

[@Psykolord1989]

The factors of 2020 are (2)(1010) = (2)(2)(505) = (2)(2)(5)(101) So for 2020 to be the product, as a set the numbers of our logicians must be one of these pairs: (1, 2020) (2, 1010), (4, 505), (5, 404), (10, 202), (20, 101), (101, 20), (202, 10), (404, 5), (505, 4), (1010, 2), (2020, 1) We have to narrow down our choices from this. Why? Because there are roughly, let me see here...2019 ways (if we care about order, closer to 1000 if we don't) for us to add two positive integers and get 2020. Fortunately, Alzim's very first statement tells us that we're working with one of the 12 pairs listed above. So Alzim must have given either 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, or 1010 as his number (2020 doesn't count because 0 is not greater than 0, so 2020 added to a positive number cannot equal 2020). So, since we can dispose of 2020 as an option, we've narrowed our pairs down to (2, 1010), (4, 505), (5, 404), (10, 202), (20, 101), (101, 20), (202, 10), (404, 5), (505, 4), (1010, 2). That said...try to sum up any of these pairs. You'll never reach 2020! In fact, the highest you can get is 1012.There is only one factor of 2020 that can be added to a factor of 2020 and sum to 2020; 1010. Thus, Era must be holding 1010, if he doesn't know after Alzim's initial answer. (For those wondering: Alzim must be holding either 2 or 1010, and we don't know which)

[@mohitrawat5225]

Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@gled9880]

It is not necessary factorise 2020 to solve this, beyond dividing by 2. We know, as soon as the result is announced to be 2020 that both must have chosen a number less than or equal to 2020 since if either was greater then both their sum and product would be greater. When Alzim gives his first answer, we can deduce that his number must be a proper factor of 2020 because if it wasn't a factor, then he would know that 2020 was the sum and could subtract his number from it to obtain Era's, and if Alzim's number was 2020, then he would know that 2020 was the product and so he would know that Era's number had to be 1. Identical reasoning tells us, as soon as Era says "no", that his number is also a proper factor of 2020, but we can deduce more, because Era knows more. Era also knows, at the time of his answer that Alzim's number is a proper factor. If Era's number had been less than half of 1010, he would have reasoned that if 2020 was the sum, Alzim's number would have to be between 1010 and 2020, and this is impossible because there is no proper factor in this range. Era's number also cannot be greater than 1010 for the same reason Alzim's can't. Therefore Era's number is exactly 1010, and Alzim's is either 2 or 1010. This reasoning generalises. Even if we had not been told the result of the calculation, only that Mufti had told it to Alzim and Era, then after both Alzim and Era had indicated in turn that they did not know the other's number, then we could deduce that the result was even, that Era's number was exactly half its value, that Alzim now knows this, and that Alzim's number is either 2 or the same as Era's.

[@MathMule]

1. First "no" indicates that Alzim's number is one of the factors of 2020 because otherwise he would know that it is impossible to make the product of 2020 and therefore 2020 must be the sum and since he knows his number, he can easily find out Era's number. 2. Same logic applies so we know and they both know both their numbers are factors of 2020. Given this knowledge, if the there wasn't a pair of numbers that summed up to 2020 then Era would know the number on the card had to be a product and find the respective number, so there must be a two factors that summed up to 2020 which is just 1010. Therefore Era's number must be a 1010. 3. Now Alzim knows this which is why he shouts "yes!" but we actually can't know whether Alzim's number is 2 or 1010 because both fit the criteria for being a factor of 2020 and since we still don't know whether the card is the sum or product then the conclusion is Era's number: 1010, Alzim's number: 1010 or 2.

[@dakyion]

The initial knowledge of the game is (1) Two positive numbers (2) their sum or their product is 2020 The first player will not be able to determine the other's player number if both equations are satisfied but we know that addition is always satisfied so he has to check that his number is a factor of 2020, if so he will say NO. The second player has an additional information (3) first player has factor of 2020 As before, for the second player to not be able to determine the first player's number, three equations are to be satisficed. (addition, multiplication and the first player's number is a factor) first he must have a factor of 2020 second the sum of the factors is 2020 also if factors are different then their sum is not 2020, otherwise it is. Answering by NO will make it possible for the first player to conclude that the second player's number is 1010.

[@zahidhussain251]

I worked out Era's number, but did not realise that there could still be two possibilities for Alzim's number.

[@AceBLify]

I think I have a bit of an easier way, which I found quite quickly. I just thought that when azim says no his number must be equal to or lower than 1010, otherwise he would get eras number by subtracting his number from 2020. Same is true for eram, because he says no. So both numbers must be equal to or lower than 1010. So either it is an addition of 1010 + 1010 or a Multiplikation. If era has a number that multiplikates to get to 2020 but isn’t 1010, he would know azims answer as there is only one option respectively. Thus the only way for era to not know is that his number is 1010 so that azim could have 2 or 1010. Good explanation though. Loved the challenge

[@jgoog7442]

Easier Solution: Step 1 - Alzim's 1st answer - Alzim must have a factor of 2020 otherwise he would know Era's number added to his is 2020 Example, if Alzim had 2018 he would know Era had 2 Alzim also cannot have 2020 because Era would have to have 1 in that case Alzim has 11 factors to choose from that could also be part of a sum so he cannot solve Step 2 - Era's answer - Era knows that Alzim must have 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, or 1010 so Alzim could not solve Era must have a factor of 2020 but not 2020 for the same reasons as above But also Era must have a number that can be both a factor of 2020 or part of a sum so he can only have 1010 Example, Era cannot have 4 because he would then know Alzim has 505 Therefore Era cannot know if Alzim has a 2 (most likely) or by luck picked the same number as Era (1010) so he cannot solve Step 3 - Alzim's 2nd answer - Alzim now knows all of the above so Era must have 1010 Alzim probably has 2 because both having the same number is unlikely but he could also have 1010

[@heco.]

these logical problem always gets me

[@dj_laundry_list]

The contrapositive being that illogical problems don't get you. I get you. Therefore I am a logical problem.

[@원펀맨-b6e]

If they chose number in any real number>0, Alzim and Era must be 1,2019 If Alzim chose number 1, after checking card 2020 he don't know if Era's num is 2019 or 2020. If it is 2020, Era will notice Alzim is 1 because it can't be zero. Therefore, it must be 2019 and Alzim would say "yes" who didn't miss it.

[@azuarc]

Looked at the thumbnail before watching and puzzled it out correctly.

[@bastianrevazov7425]

12:24 i love the way he says "YES!"

[@DrCeeVee]

I approached the problem in a slightly different way than in the video. The fact that Alzim doesn't know Era's number (1st reply) means that Alzim's number must be a factor of 2020. Otherwise, he would be certain that 2020 = x + y, and hence he could solve for Era's number (y = 2020 - x). Furthermore, Alzim's number can't be 2020, because then he would be certain that 2020 = x * y (since 0 is excluded), and hence, again, he could conclude that Era's number is 1 (y = 2020 / x = 2020 / 2020 = 1). Therefore, Alzim's number must be a factor of 2020 that is at most 1010. Era now knows that Alzim's number is one of 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010. For the same reasons, Era's number must also be one of these numbers. But then, all of these candidate numbers, except 1010, can only satisfy the product equation 2020 = x * y, and not the sum equation 2020 = x + y, since for any one or two numbers less than 1010, the sum x + y < 2020. Therefore, if Era's number was less than 1010, he would be able to use the product equation and solve for Alzim's number (x = 2020 / y). Only the number 1010 satisfies BOTH the sum (x = 2020 - 1010 = 1010) and the product (x = 2020 / 1010 = 2) equations. Therefore, if Era's number is 1010, he cannot determine Alzim's number (it can be either 1010 or 2), so he answers "no". Alzim can now conclude with absolute certainty that Era's number is 1010.

[@NatoSkato]

From the first no Alzims possible numbers are reduced to the divisors of 2020. From the second no we understand that Era is faced with some ambiguity. If his number added to any of Alzim's possible numbers totaled 2020 without his number multiplying with another to give 2020, he would know what Alzims number is. Since he doesn't, it means that Era's number both adds with one of the divisors of 2020 to total 2020 and multiplies with another. The only divisor which fits this description is 1010.

[@asaleemeadows]

I'm going with 101 and 20, because of the factors of 2020, if A had 101, that means that E could have had 20 or 1919. If E had 20, he would have known that A had 101. Since he didn't know, it meant he had to have 20. All the other factors, 1,2,4,5,10, and 1010, would have given E either 1 option only, and he could have deduced A's value, except for 20, 2000 and 101,1919. Since 20 would have been either 2000, or 101, E would have been to figure it out. So that excluded 20, which left 101 and 1919. And if A had 1919, he would have been able to know that in the first round. So E not knowing let A cross that off, and it left 20. Now to continue at 1:16 to see if I'm right.

[@asaleemeadows]

crap. lol. I had 2 x 1010 and 1010 + 1010 in my list, but I crossed that off for some reason. Now I'm trying to see where I went wrong with 101,20. :(

[@MarkV-f3q]

Mufti: *read the question* Alzim: "No." Era: "No." Alzim: "Yes." Spectator: "OOOOOOO" 🔥🔥🔥👏👏👏

[@herumuharman6305]

TLDR; 2 people, A and B; 2 numbers, a and b. From A's perspective, there are 2 possible b, b-sum or b-prod. From B's perspective, it's either a-sum or a-prod. First question. A don't know B's number, so "a" has to be factor of 2020, because if it's not, A can easily guess that b can only be b-sum. Second question. B don't know A's number. Same logic applies for B, b is a factor. But because of the first question, now they both know that both of their number is a factor of 2020. After second question we know that a and b is factors of 2020, but because B still can't guess A's number despite of this information, that also means that both a-sum and a-prod are factors of 2020. a-prod has always been a factor of 2020, so it's a-sum that is ambiguous. So now we need a number for "b" where the difference from 2020 is also a factor of 2020. Only 1010 satisfy that.

[@luisfilipe2023]

I can’t believe I didn’t figure out the very easy solution to this problem. Math truly hinges on one not being lazy and fearful

[@RoderickEtheria]

1)If Alzim's number is 2020, then he would know Era's number is 1. If Alzim's number is not a factor of 2020, then he would know Era's number was an addition. 2020=101×2×2×5 Possible numbers for Alzim are 1, 2, 4, 5, 10, 20, 202, 404, 505, 1010. Era could instantly deduce the same thing about Alzim without Alzim's info, but still cannot determine whether it is addition or multiplication. He knows Alzim has 2 or 1010, and Era has 1010.

[@matrenitski]

I solved it easier. Azim If x>1010 => yes If X not in [1,2,4...1010] => yes So X in [1,2,4...505,1010] => No Era: Same logic, y is in [1,2,4..1010] Any number below 1010 would give a clear yes. Only y=1010 leaves two options x=2 or x=1010 Easy-peasy 😅

[@humberkger]

I haven't watched the whole video yet, but I already got the answer (this maybe went horribly wrong) Premise 1: Asking Alzim, but he replied with yes. It means that Alzim's number is a non-factor of 2020, and he already knew that his number and Era's must be added to form 2020, not multiplied. So he just calculate 2020 minus his number and it's Era's number. OR both of them just happened to pick 1010. Premise 2: Asking Alzim, but he replied with no. Then, asking Era, but he replied with yes. It means that Alzim's number is a factor of 2020, which he is conflicted, whether Era's number is an additive or multiplicative to his number to be equal to 2020. (Little example, Alzim chose 505, he is conflicted whether Era wrote 1515 (add 505, equal 2020) or 4 (multiply by 505, equal 2020)) But, since Era answered yes, which means Era knows exactly Alzim's number, which turns out to be the same case as Premise 1, but with Era this time; Era's number must be a non-factor of 2020, and he knew exactly what Alzim's number is. Premise 3: Asking Alzim, but he replied with no. Then, asking Era, but he replied with no. This, continuing from Premise 2, means that Alzim's number is a factor of 2020, which he is conflicted (as the reason explained on Premise 2). Then, Era did also reply with no, which means, he is also (somehow) getting conflicted with the same reason. It turns out that Era's number is also a factor of 2020, same as Alzim's. Then, we can conclude that both numbers must be 1010 since both are the sum and factor of 2020. [Assumption] So since Era somehow also answering no, to the 2nd question, we can assume that they don't know each other's answer to the question. If Era heard/knew that Alzim said no to the question on the 1st time, he should answered with yes on the 2nd question, since he already knew whether Alzim's number is a factor of 2020 or not (which it's not, since Alzim answered no). Also, on the Premise 3, logically, should Alzim be asked whether he knows Era's number, he should always answered with Yes, since 1010 ALWAYS get paired with 1010 and conversely should happen when Era's got asked the first question, which still follow under Premise 1, 2020-1010 is 1010.

[@Khantia]

I couldn't quite think of an exact number, but I thought of several possible pairs: (2, 1010), (4, 505), (5, 404), (10, 202), (20, 101).

[@stanimir5F]

On top of the nice math problem I appreciate the fact that you choose as "random" numbers 69 and 420 :D GIGACHAD move! :D

[@AnantSiwach]

i solved this riddle in first try even I am amazed, I solved it from era's side instead of saying no he should've know if he. was truly a great logician

[@nicholasimholte7359]

Wow I got one! This seemed like it was on the easier side. Good explanation

[@Jon60987]

Awesome problem with an entertaining solution. Thanks for this video :)

[@novacorponline]

Something I do not get here is, why Era can't solve the puzzle on his turn. Alzim gains new information on his second turn which lets him solve it, but Era has *the same information* during his turn.

[@1votdopeningfan138]

because while both have access to the same pool of information, this information boils down to - era has 1010 - alzim has 1010 or 2 both of them know this, your right. the problem for era is that he needs to know what alzim has, while alzim has to know what era has. while both have access to the same information there, its more useful to one of them than the other

[@heinmic17]

When the first question is asked, if alzim has 1010, alzim is considering whether era has 1010 or 2, but if alzim has 2, alzim is considering whether era has 1010 or 2018. In either instance, alzims answer of "no" is valid and provides era with no information about alzims number that era did not already know before the question was asked, because era must have already assumed alzim had either 1010 or 2.

[@GregTom2]

Let's see. Alzim said no because he knows that his number could either be be a factor of 2020 (2*2*5*101), or add to 2020. Era said no because, knowing that Alzim's number is 2, 5 or 101 or a product thereof, his number could both add to 2020, or multiply to it. This means his number is a factor of 2020 that can be added to another factor of 2020 and result to 2020. There is only one such number: 1010. Alzim says yes, because he knows Era's number is 1010. Alzim's number is either 2 or (by unbelievable coincidence) 1010.

[@mohitrawat5225]

Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@zakyzigzag]

Hello fellow Indonesians who might stumble across this video! I'm from Nganjuk LOL

[@PekoraMamiCouncilHK]

I tried to simplify the solution but explain in 2 directions First, made 2 equations and we have 1) E+A=2020 2) E•A=2020 But only ONE of them could fit the number om the card. Next we have to analyze the "No, No, Yes" description from both sides. We have 1N because no one knows it is either the sum or product of the list. Before the 2nd person's reply, we should know that the 1st person didn't knows which equation could obtain 2020 AND which number to fit EITHER equations. He also can't conclude the combination to obtain 2020 by adding as well. Since the 1st answer could not provided ANY usable information other then these. The 2nd person should not able to make any conclusion. But we should realize that only the nubmer is ONLY able obtained by adding at this moment. Because only "adding"/ equation 1) will have such many possibilities to confuse both person. We have only 1010+1010=2020 Because only this combination can fit the situation of "Yes as the third reply" The logic might be "reversed" But only this can fit the case. Since if ANY person didn't picked 1010. Then they will only repeat "No" since the 3rd reply. Both person will never have any idea and produce any usefull information by Yes or No. Cheat: It must not be obtained by multipling/ equation 2). Since they MUST know their own decision AND there is only ONE possible number picked by the other person. (Only 2 numbers in each combination). If it is obtained by multipling/equation 1). We should instantly have a Yes as 1st reply. As a audience, we should not know which number picked from a specific person. Since we still have a problem "a list of 12 combinations presented in 11:51 " Finally we have an additional MUST-HAVE equation for the 3Y. E=A 2020=2A=2E Then A=E=1010

[@PekoraMamiCouncilHK]

I take an hour to obtian the conclusion and typing job At least, I am sorry for my poor English. I found one thing that made the question so confusing or unfair to someonebody. Because the remaining card was still HIDDEN. It could not prove they made the correct guessing or NOT. Even if E=1010 and A=2. We will have 2 cards with 1012 and 2020. I can prove my opinion with arround 2020/2=1010 combinations to challenge the "provided answer" In my opinion, this puzzle is not solvable since we lack of a vital information.

[@HarishKini]

fairly simply puzzle made complicated by your explanation

[@Drteslacoiler]

My path to answering to this questiıon is simpler IMO. STEP 1) Before being asked; ALZIM already knows that if his number is not an element of the set (2, 4, 5, 10, 20, 101, 202, 404, 505, 1010) then multiplication must be automatically eliminated thus he can subtract his number from 2020 and know the number of ERA. But he says NO. This means that his number is an element of the set (2, 4, 5, 10, 20, 101, 202, 404, 505, 1010). => So we know that ALZIM's number is of one the numbers in the set (2, 4, 5, 10, 20, 101, 202, 404, 505, 1010).and ERA's number is either '2020 minus one of the numbers in the set' or '2020 divided by one of the numbers in the set'. STEP 2) The first logic also applies to ERA. Since he says NO his number must be one of the numbers in the set (2, 4, 5, 10, 20, 101, 202, 404, 505, 1010). Furthermore, the only number to create a condition that he can not exactly know ALZIMs number is 1010. For example if his number was 404 then he could conclude that ALZIMs number has to be 5 (addition is not a possibility for 404), and so on. But in the case of 1010, both addition and multiplication are possibilities since ALZIM can be either 2 or 1010. => So we know that ERA's number is 1010 otherwise he would not answer as NO STEP 3) Based on the answer of ERA, ALZIM concludes that his number is 1010 and says YES. But we can not figure out his number since it can be either 1010 or 2

[@MiccaPhone]

I stopped at 1:24 and came up with this solution: Azim's number is 2. Era's number is 1010. Azim says "No I dunno" because Era's number could be 1010 or 2018 from Azim's point of view.. Era says "No I dunno" because Azim's number could be 2 or 1010 from Era's point of view, and in both cases Azim couldn't know the other number, so Azim's answer "No" does not give Era any extra information. Now Azim says "Yes, now I know", because Era's number cannot be 2018. Because if Era's number were 2018, Era would know that Azim's number can only be 2. Now Era understands that Azim's number cannot be 1010 because then Azim couldn't have answered "yes" after Era's "No". So Era now also says "Yes, now I know, too" and knows that Azim's number must be 2. Another solution is not possible, because if Azim's number is greater than 2 (like e.g. A=4 or 5) Era's number is smaller than 1010 (like e.g. E=505 or 404). But then, after Azim's first "No", Era would know that Azims number A must be "A=2020/E", because in case of A=2020-E, which is greater than 1010, Azim would have answered with a solid "Yes" from the start. So (A, E)=(2, 1010) is the only solution.

[@MiccaPhone]

Just to add: (A, E)=(1, 2020) is no solution, because Era would know from the start that A can only be 1 since A=0 is not a positive number.

[@MiccaPhone]

Obviously I stand corrected, since (1010, 1010) is another solution!

[@darrenhundt]

the funniest part of this video is how all those "twenty"s hammer home how Presh never elides letter sounds. saying "twenty" instead of "twenny" :)

[@ilchad8016]

2020=2*1010 so Era It must have chosen either 2 or 1010. If we have 2 "No" and 1 "Yes" "No" Is greater than "Yes", but, two denials gives a statement, and "Yes" Is also a statement, so two statements gives another statement. With numbers we have 1+1=1 in boolean algebra. This means they have through the same number, but which one? Well, if we give to "Yes" the number 1010, and "No" the number 2, "No"*2+"Yes"="Yes"+"Yes"= 1010+1010=2020, but this mean both were thinking the same number, which Is difficult. We sit still on this problem: know which number Era was thinking.

[@handanyldzhan9232]

The number is 2020. Let their numbers be A and E. From Alzim's PoV, Era's number can 2020-A or 2020/A. Both options are possible, because his number is a positive number less than 2020, and it divides 2020. From Era's PoV, his own number has the same restrictions (2020-E or 2020/E), and he now knows Alzim has the same. Then Alzim figures it out. He now knows Era's number isn't only a positive integer (and less than 2020 either way unless A=1), it also divides it (and is less than it in any case). If 2020-A doesn't divide 2020, the only choice left to him is 2020/A. But E didn't figure it out, so 2020-E divides 2020. If A=2, E=1010.

[@UserSOF0]

your videos had a great style why bring ai to it

[@pbenikovszky1]

I would not call this impossible :D Here is a pretty easy solution to think through :) So we know that 2020 is either the sum or the product, because both says no we know that both has a factor of 2020 (otherwise any of them could figure out the other number). So we need to find a factor that can be in a sum and also in a product to get 2020 with another factor. As 2020 is even the most obvious choice is the half, so 1010 as 1010*2=2020 and 1010+1010 =2020

[@robinlindgren6429]

each participant has a number, A has x and B has y such that either 2020 is the sum or the product of x and y. with this information that means that the other persons number can only be one of two possibilities. for example if I know x, then y is either 2020-x or 2020/x. there are some values of x that would exclude one of those possibilities, namely that x would be 2020 (which would force y to be 1) or x would have a prime factor other than 2, 5 or 101 (and no more than 2 factors of 2 or 1 factor of 5 or 101), since having any other prime factor combination would cause 2020/x to not be a whole number, thus concluding that y is 2020-x. A is asked; do you know Bs number? and A answers no. that means the above criteria that would exclude one option is not fulfilled, so As number must be either 2, 4, 5, 10, 20, 101, 202, 404, 505 or 1010, which are the only numbers where B's number could not be concluded. B is asked, do you know As number? and B answers no. that means B's number can be multiplied by one of As possible numbers to get 2020 and can be added by a different one of A's possible numbers to also get 2020. we can therefore generate our possible candidates by subtracting the possible x from 2020. subtracting a small value from 2020 would produce a number that is too big to be a factor of 2020, in fact even subtracting the second-largest number (505) would still produce a number that is far too large to be a factor of 2020, therefore we must have that Bs number is 2020-1010=1010, since that is the only value that can both be multiplied by a number on the list (2) and added to a number on the list (1010) to produce 2020. so y=1010. Bs number is 1010 and As number is either 2 or 1010.

[@jeanf6295]

Let A be Alzim's number and E be Era's number. We know that A and E divide 2020 as otherwise, both characters would know 2020 = A+E. If 2020 = A+E then A divides A+E and thus A divides E. Likewise E divides A+E and thus E divides A. Therefore A = E = 1010 Era is unable to conclude because if 2020 = A*E then A = 2 and E = 1010 is also a valid solution. Thus E = 1010, and A = 2 or 1010.

[@brokendoll3368]

Okay. If the number 2020 is the sum of both numbers, than one of the two numbers must be between 1010 and 2019 (inclusive). If however 2020 is the product, any number from 1011 and 2019 is out. Had Alzim any of those numbers, he would know Era's number right away. So Alzim's number is between 1 and 1010 (can't be 2020 either because that only leaves the product with 1). Era can deduct that. If Era's number is above 1010, he would know it is a sum and know the answer. If below 1010, it must be the product instead. So the only option remaining is 1010, leaving both 1010 and 2 as legit possibilities for Alzim.

[@barttemolder3405]

Exactly. The only tricky bit is realizing that though we do know Era's number, Alzim's can still be either 2 or 1010. In fact we could use any even number not just 2020. The second player will always have half that number if the game proceeds to the 3rd question.

[@mohitrawat5225]

​@@barttemolder3405Alzim's number is also 1010. Here's the reason - There can be 2 possible answers for Era 2 or 1010. If Era's number is 2 then he would have known that Alzim number would be either 2018 or 1010. In that case he would have released that if Alzim number is 2018, Alzim would have known Era number cause you can't get to 2020 by multiplying 2018 with a whole number the only way is to add 2. But Era said no. So it means Era number is 1010 cause in that case he woupd have been confused.

[@barttemolder3405]

@@mohitrawat5225Both 2 and 1010 are possible. If Alzims's number is 2 he knows that Era has either 2018 or 1010, but not which one yet. If Era has 2018 he'd know the answer so as he doesn't Alzim realizes he must have 1010. Alzim can have either 2 or 1010. Era has no way to know even after Alzim revealed knowing Era's number.

[@rohitdakhane9181]

​@@barttemolder3405 yup

[@zugzwang2007]

Good puzzle, but not impossible! There is a much less laborious way to the answer (though with elements in common). The reason Alzim doesn't know Era's number y at stage 1 is that his own number x is a factor of 2020. Parallel reasoning helps Era to get further, however. As he knows not only that Alzim's number x is a factor of 2020 but that it is either the case that xy=2020 or that x+y=2020. The problem for Era is that he doesn't know which. But he does know his own number y. There is only one factor of 2020 that y could be, and still satisfy either equation, namely 1010, so Alzim knows that 1010=y. What Era doesn't know is, more precisely, whether x=2 or x=1010. But Alzim knows which of these it is. And whether it is the sum or the product equation that is true of x and y, and makes the "or" proposition true. And he knows, in either case, that y=1010. And that is what we know about Alzim and his number.

[@ChirpingMatt]

Thought process before seeing the answer: We know that 2020 is either the sum or product of both numbers, which must be positive integers. If Alzim had a number that was NOT a factor of 2020, he would know it cannot be the product, must be the sum, and could quickly work out Era's number. Because Alzim didn't know, his number MUST be a factor of 2020. Era now knows that Alzim has a factor of 2020, bit still cannot determine Alzim's exact number. In order for this to be true, two things have to also be true: 1) Era ALSO has a factor of 2020, otherwise he would know it must be the sum and can work out Era's number. 2) Era has specifically a factor where 2020 can still potentially be the SUM of the two numbers. However, there is only one way a pair of (non-distinct) positive factors of a target number can sum to the target number, which is if BOTH are exactly half of the target number. (Informal proof: If two positive integers are summed, at least one must be greater than or equal to half of the sum. The only possible factor bigger than exactly half, is the target number itself. But adding any other factor to it necessarily gets you a total BIGGER than the target number.) Thus, Era MUST have exactly 1010, since that is the only factor he could have where it is still unclear if 2020 is the sum or product. Interestingly, since Alzim can conclude this regardless of which possibility he had, we still don't actually know if 2020 is the sum or product. It is still possible for Alzim to have EITHER 2 or 1010.

[@yusaldyhikmara4109]

On the original pdf, Alzim answer no first, then era answer no. But there is no Yes answer afterward from Alzim. Where do you get that last Alzim's answer first? And does it impact the logic still?

[@abduleusha1601]

I still couldn't get the point that why (2020-y)|2020 is taken. Can someone pls explain?

[@rajafarhat5993]

I can safely say that the way you solved it is not ideal

[@YaacovIland]

That result can be generalized.

[@muskyoxes]

He must get a lot of "I don't get it" comments, because he saturates his answers with microsteps and fussy repetition

[@informationgiant7697]

I confirm what you say❤

[@bharathram7245]

How this channel growing with the wisdom! 🎉

[@jefft5854]

This one is rather easy, for once.

[@SilentTM]

especially after seeing the solution, huh?

[@gamefacierglitches]

Don't _need_ a chart for the last bit, as it can also be solved with logic. You need any two factors of 2020 that add up to 2020. Since 1×2020 is obviously incorrect, you are looking for two numbers

[@DmitryNetsev]

They cannot find answer quickly, because both of them pick number from small pool of 2020 divisors : 2, 4, 5, 10, 20, 101, 202, 404, 505 and 1010 (1 and 2020 cannot be, because 2020 + positive number > 2020). But Era knew, that Alzim don't know answer - what's mean, that Era knew, that Alzim pick number from same pool, so Era still don't know answer only, if his number is 1010. But if they both pick 1010, no one of them can release another number. So answer is: Alzim's number is 2, Era's number is 1010.

[@muncie1742]

Loved this logic problem... took me about 10 minutes with note taking but i figured it out.

[@Arkayjiya]

I did the opposite, I knew from the beginning that the solution was almost certainly 1010 since that's the only number that has a counterpart both for addition and multiplication which is why there's two uncertain answers in a row before we get to the answer, but I couldn't bother trying to actually go through the logic, so I technically didn't figure it out before watching the video. On top of not having proven the answer, since I didn't do the work, that means I had no way to know if the first number was also 1010 or 2. Edit: apparently going through the reasoning rigorously also doesn't let you know whether the first number is 2 or 1010 so I guess I'm not alone here xD

[@TK669]

420 and 69 are not random numbers you sneaky mathematician

[@tvrizzle1]

I figured it out before watching, glad I tried it!

[@Primalmoon]

I worked on this from just the thumbnail, which didn't call out the "greater than 0" restriction. I still made the assumption of both numbers being from the natural numbers, but I allowed for x and y >= 0 It needed a bit more thinking because of the possibility that someone could have their number be exactly 2020 and still not know the other person's number, but the "No, No, Yes" sequence still forced the same result of 1010 for Era. Alzim saying "No" implies x != 0 and x|2020, but x=2020 is still valid... Era saying "No" implies y != 0, y!=1, and y|2020, and forces x,y

[@MarcusAndersonsBlog]

This is not a puzzle in maths or logic. It's an exercise in seeing though the heavily disguised equality at work, which is: x+x = 2*x for all whole numbers x, and that one of the numbers must always be 2. This cannot actually play out as it is described unless one of the players, the victim (Alzim), is not in on the trick - the one that doesn't pick the number 2. Because it is so difficult to figure out, the victim is actually very unlikely to complete the last part of the deception by announcing they know the other person's (Era's) number, and this completely ruins the party trick in real life. This is because the other players must all lie - they must already know the victim's number is always the number given divided by 2. Much the same trick can be played for the number 3 using 3 participants and the equality x+x+x = 3*x. In general, the equality of SUM(x1, x2, x3, ... xn) = n*x is the simple strategy at work, where y will always be equal to n. No need for logic or complex maths. Its basic. However all this does make for a ripping puzzle, but it also involves deception, no matter how it's put. That is not fair in a timed maths contest because anyone who knows this party trick will be significantly advantaged, and every one else significantly hindered.

[@shykitten55]

Ok, only started watching it. This is like the puzzle way way WAY back where to people and a birthday. And I'm sure I've seen very similar "flavours" of this elsewhere too.

[@oralon99]

I will try my luck, Alzim says no which mean thay he doesn't know if its a product or a sum, that means that his number divide 2020. Than Era says no, knowing that alzim number divide 2020. That means that Era number also divide 2020, further more it means that summing this number with enother divider can get you 2020, and thw only number that does that is 1010. So now Alzim knows Era number is 1010 but his number is either 2 or 1010.

[@oka-yoke]

Alzim= 101 Era=20 Lets assume 2020 is the production There could be options for Alzim; A= 1 2 4 5 10 20 101 202 404 505 1010 2020 E= 2020 1010 505 404 202 101 20 10 5 4 2 1 If it was 1 for A, E would be either 2020 or 1019 if it was sum. But Era could guess both of them at first attempt. If E doesnt know at first attempt then it cannot be 1 then we eliminate 1. Lets select another number as a number which is prime number except 2. It is 101. If A selects 101 and if E was 1919 and it was a sum then E could guess at first attempt but if he cannot guess at first attempt then it is a production and E is 20.

[@ab-mc2nq]

Ok so before seeing the answer: If Alzim's number was not a factor of 2020, then he could conclude that it was the sum card and do subtraction to find Era's number. Era knows this too, so in addition to also being a factor of 2020 for the same reason as Alzim's number, Era's number must also sum with a factor of 2020 to make 2020, as well as multiply with a factor of 2020 to make 2020. Era's number is 1010, as Alzim could have either picked 2 or also 1010

[@darcash1738]

No, No so both must have factors of 2020, or else it would be an immediate Yes for adding. More on the second No: Since E knows A must be a factor, he must have a number which is a factor and can also add to 2020. 101*5*2^2 is the prime fact. so the only one where that's possible to add to another one and be able to multiply to it is 1010. which is how A knows.

[@TheRealJman87]

Those AI hands are something else. I wonder whose art was copied without their permission to train the AI