Impossible Logic Puzzle from Indonesia!
ฝัง
- เผยแพร่เมื่อ 9 ก.ย. 2024
- This is one of the hardest logic puzzles I have ever come across.
Translated and slightly re-worded from a 2018 math competition in Indonesia by Gadjah Mada University. Question 15 in this pdf:
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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.”
The third says without sugar
@@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.
@@NathanSimonGottemer what a twist
@NathanSimonGottemer I caught the "all" in the first part immediately, but had to read the second part twice to notice the "any". Great joke!
@@cynicviper ehe thanks. I do love a nerdy joke.
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.
There are only 3 conclusions you found, and (4,550) satisfies all 3 of them. And btw there isn't only one solution.
@@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
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".
That's a really nice way to solve it!
@@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.
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.
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.
@@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.
@@aaronbredon2948 Ah, gotcha. I was thinking about Era's number, not Alzim's. Makes sense
Era should've been able to guess it
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".
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.
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.
My logic was the same. ]
But still why said yes era on second round? Is still between 2 and 1010
@@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.
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.
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.
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?"
@@JohnDoe-ti2np i don't think that holds true if the number is not divisible by 2
@@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.
@@JohnDoe-ti2np yea that's true! Cool stuff
@@JohnDoe-ti2np Or you could turn it into a _real_ trick question and just ask "Does Alzim know what Era's number is?"
Let me just randomly pick some numbers to show you why they don't work. I don't know...how about 420 and 69?
How about no?
meme math
That's exciting...
You couldn't say 42, because that's the answer
nice
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.
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
@@nahommerk9493 Presh is wrong. Both numbers are 1010. Alzim's number cannot be 2
@@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)
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
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!
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.
@@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.
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.
My mind sleeps while watching 😂
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.
I love Cracking The Cryptic! Two very great channels indeed.
@@guilhermeteofilocachich4892 Indeed!
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
Much cleaner than the solution in the video!
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.
@@mohitrawat5225you didn't explain why Alzim's number should be 1010 at all.
these logical problem always gets me
The contrapositive being that illogical problems don't get you. I get you. Therefore I am a logical problem.
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.
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.
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!
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)
@@user-jz7vf5iq7hAlzim'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.
@@mohitrawat5225again you just copy pasted and did not explain why Alzim's number must be 1010. You just talk about Era.
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.
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!
love your videos presh but ai art is meh
True😂
True😂
True😂
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.
@@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
I can safely say that the way you solved it is not ideal
That result can be generalized.
He must get a lot of "I don't get it" comments, because he saturates his answers with microsteps and fussy repetition
I confirm what you say❤
Looked at the thumbnail before watching and puzzled it out correctly.
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.
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.
This one is rather easy, for once.
especially after seeing the solution, huh?
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.
This was a really good one. I couldn't figure it out until 8:14 into the video.
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.
y | 2020 => ty = 2020
(2020 - y) | 2020 => (t-1)y | ty => (t-1)| t
This is ONLY possible when t = 2 therefore y = 1010
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.
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.
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).
Almost right. It doesn't work when N = 4. Otherwise it does for any even integer > 4.
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.
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!
what the heck is going on with Era's hand
Era isn't real, you know? He was imagined by some generative AI in its infancy, which is still confused about hands.
Why are they Indonesian
@@bilkishchowdhury8318 because… the puzzle is from… Indonesia…?
The whole video is how Alzim thinks Era would guess his number. On the second turn Alzim already knows what number he had given so he knows X
your videos had a great style why bring ai to it
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.
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.
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
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.
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.
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)
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.
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.
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.
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.
Loved this logic problem... took me about 10 minutes with note taking but i figured it out.
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.
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.
I know this puzzle for more than 30 years from a math contest in the University and I can be quite sure that it is not Indonesian.
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
Era's number is 1010 ; Alzim's number is either 2 , or 1010 too.
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.
@@mohitrawat5225 Sorry, but no. And your explanation doesn't show that Alzim's number is 1010, it only shows that Era's number is 1010.
We can't derive Alzim's number, it's either 2 or 1010, we cannot know which. Let's look at both scenarios:
Scenario A: Alzim's number is 2 and Era's number is 1010 :
Alzim has 2 , therefore he knows Era's number is either 2018 or 1010 , but at first he doesn't yet know which, so he says "No, I don't know Era's number." Era can now conclude that Alzim's number is a divisor of 2020 , but he doesn't know if that divisor is 2 or 1010 ; since Era has 1010 , both possibilities could work to produce 2020. Therefore, Era also says "No, I don't know Alzim's number." At that point, Alzim knows that Era cannot have 2018 (otherwise Era would have said "Yes"), so Era's number must be 1010; so now Alzim answers "Yes, I know Era's number."
Scenario B: Alzim's number is 1010 and Era's number is 1010 :
Alzim has 1010 , therefore he knows Era's number is either 2 or 1010 , but at first he doesn't yet know which, so he says "No, I don't know Era's number." Era can now conclude that Alzim's number is a divisor of 2020 , but he doesn't know if that divisor is 2 or 1010 ; since Era has 1010 , both possibilities could work to produce 2020. Therefore, Era also says "No, I don't know Alzim's number." At that point, Alzim knows that Era cannot have 2 (otherwise Era would have said "Yes", because Era would have concluded that Alzim doesn't have 2018), so Era's number must be 1010; so now Alzim answers "Yes, I know Era's number."
See, both scenarios work out; so Alzim's number could be either 2 or 1010 .
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.
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)
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.
Somehow this seemed super easy to me. took me no more than 5mins to figure it out. unlike other problems on this channel :)
Lovely cameo by my good friend, 420.
And my bff, 69.
Wow I got one! This seemed like it was on the easier side. Good explanation
On top of the nice math problem I appreciate the fact that you choose as "random" numbers 69 and 420 :D
GIGACHAD move! :D
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.
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.
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?
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
I arrived at the solution in another easier way: Al say he don t know the nunb, it means it's a number smaller then 1010, because if it was bigger, he would know the other number because the paper is the sum and not the prod, becahse otherwise it would be much bigger. It can't be smaller then 1010 because otherwise he would have known the other number, because the second no. If it's smaller then Era would have a bigger number then 1010 and he would have know the other number by the same reasoning. Then it's 1010. It's not easy to explain why i pick 1010 with only a comment
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.
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.
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.
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.
I love how this channel isn’t above slipping in 420 and 69 😂
Nice
I solved it! *patting myself on the shoulder*
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.
Actually it's much easier. If 2020 is not divisible by Azims number Azim can answer in the very first turn. Since Azim answered no it can not be a divisor of 2020, thus it has to be the sum.
It was exciting . I started just like in this video , but only up to 7.15 .
Well... I'm happy I solved this one.
Now start reading the smarter solutions of others🙂
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.
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.
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. :(
There is a logical feedback loop in this puzzle...
Because if Alzim is able to figure the number out by Era saying the second "No" - then surely, Era would be able to figure out the number _in the case of him saying no_ - so even before saying no, just by knowing that his answer WOULD be "no", he knows the number. But if he knows the number, he has to say "yes", which invalidates the entire reasoning, which makes it that he has to say "no".
It's an endless loop of the "what if you travelled back in time and killed your own father before you were born" type...
I don't see how this works. Era says no because he doesn't know if Alzim's number is 2 or 1010. Knowing that he will say "No" doesn't change this uncertainty for Era.
It's a problem of who acts first
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.
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.
@@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.
@@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.
@@barttemolder3405 yup
I sat with my wife for 5 mins and we deduced the following:
1. If either have a number that is not a factor of 2020, they would know 2020 is a sum and answer "yes"
-->Therefore both men have factors of 2020
2. If both have factors of 2020, then the second would immediately know the first persons number in all cases except 1010.
-->Therefore the second person has 1010.
This is where we got stuck, we couldn't figure out how to figure out what the first guy has as it could be 2 or 1010.
Turns out this is the answer tho lol.
But the highlight is "Era doesn't know Alzim's number until the end😃"
fairly simply puzzle made complicated by your explanation
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 😅
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.👏👏
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
But Era knows that Alzim knows that the card must be the product of the numbers, so he can just do 2020/y = x and Era would know Alzims number. Therefore, Era is not a perfect logician.
How this channel growing with the wisdom! 🎉
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.
I rarely get these, but the no, no, yes sequence led me to the solution in less than a minute and without factoring, equations, etc. Go figure.
So, neither of each person's number is 420, 69, nor 404.
The reality is cruel
i like how you picked 420 and 69 as example numbers
Actually Alzim would have had to have chosen 2, because if he had chosen 1010 Era could have still chosen either 1010 or 2 still. If Alzim had said no a second time, Era would have said yes and known Alzim had chosen 1010.
It took me 10 minutes and 10 seconds to figure it out.
I'm just happy I was able to figure it out...
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.
I solved this while listening to ELO’s Sweet talking woman. They granted me the strength necessary.
If Era was the perfect logician he is, he would have saw that if he said no, he would lose the game. So the best move for him is to guess a number either 2 or 2010, that he still has 50% chance to win.
Lose what game?
There is no prize to be won. Maybe the person who figures out the other person's number first will be assigned by the Mufti to some tedious job/task.
But if Era number were 2, then he would know that the fact that Alzim doesn't know his number, then He would conclude that he has the number 1010, so Era number needs to be 1010 too
It seems to me that we can deduce Alzim's number as well as Era's. If Alzim's number was 1010, then he wouldn't know whether Era's was 2 or 1010. The only way he can know Era's number is if his number is 2.
Let Alzim have 1010. He says no in first round because Era can have 1010 or 2. But since Era says no, Era must have 1010. If Era had 2 he would know Alzim must have 1010, because Alzim cannot have 2018 since he didn't answer yes in the first round.
I think I’m starting to think like Presh Talwalkar because I opened up a Google doc and pretty much wrote down my logic to the answer before watching for the answer, and it went almost exactly like Presh’s did. I know that there were more shortcut methods possible to solve this, but I had to write it all out for the sake of proper proving.
I started out by making a table of A and E and their combinations of odd and even and whether the resulting sum or product were odd or even. If both numbers are even, you cant tell whether it is a sum or product. At that point, I inferred they were both even and the only unique even number is 2 because it is prime. I then concluded the other number was even. Not rigorous and the wrong answer but then I only spent 10 minutes on it.
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" :)
wasnt Era half-lying when he said he doesn't know Alzim's number(s), since he knew that it could be either 2 or 1010.
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.
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.
I saw this right away. I'm shocked. This never happens.
I had a nap and a green tea. Maybe that's why.
I don’t know about green tea that’s interesting but a nap definitely enhances cognitive abilities. I need to do it!
@@captainhd9741 I'm trying to explain an unusual event in my life. :)