# Can You Find the Number?

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Going back to Jack and the range.

On 2/14/2019 at 4:05 AM, Umurbaba said:

Jack says: I dont know all the digits﻿﻿ but James﻿ ﻿﻿doesn't either.

So I do not know all the digits implies he knows at least one of them.

The only way he can be certain of at least one digit is if it is a 9 since then the range is 9

There are two ways for the range to be 8.

If the range is 9 then another digit must be a zero.

This is fine for his comment about James since ther is a zero the product is zero and there are many ways for the product to be zero.

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56 minutes ago, studiot said:

Going back to Jack and the range.

So I do not know all the digits implies he knows at least one of them.

I took this to mean that Jack does not know the values of all the digits. Possibly none, a single one, or any number up to three, but not all four.

Can you explain any case, except when the range is 9, when Jack can truthfully make this statement? Because in this case he will know that $$a$$ is 9 and $$d$$ is 0.

Anyway, we agree that it follows from the entire conversation that $$a$$ is 9 and $$d$$ is 0. I certainly do not agree that it follows from ordinary english usage that Jack is in any way indicating that he knows that the range is 9.

Edited by taeto

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Jack says

"I don't know all the digits.

I have emphasised the word all since in normal English he means he knows some of them but not all 4 (unless he is deliberately concealing the fact he he doesn't know any a scenariao we are discounting).

Clearly if he knew all four he would declare the solution.and James and John would never need to speak.

Now this could be just the OP lax English, like some of the rest of the OP or it could be meaningful.

I can't tell.

So I was examining what digit or digits he could know for certain, from the information he possesses.

But we have both agreed we really need a reliable statement of then problem.

However thinking through this is quite an enjoyable exercise, do you also find it so?

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2 minutes ago, studiot said:

However thinking through this is quite an enjoyable exercise, do you also find it so?

I agree!
I'm giving this a try. For some reason I'm having a language barrier trying to write down my reasoning...

My guess is:

Spoiler

4111

I'll try to explain in a followup, not sure at all if my attempt is valid.

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52 minutes ago, studiot said:

However thinking through this is quite an enjoyable exercise, do you also find it so?

It sure is.

I am mildly disappointed though, if it really follows from the grammar of the english language that Jack's first comment actually already implies that the initial digit is 9 and the last one is 0. Something which I seemed able to deduce logically from the ensuing pieces of information.

Surely you agree that as a riddle, it would be better constructed if essential pieces of the puzzle are not revealed at the start, rather than at a point later when they can be deduced logically.

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Jack has 3 (range) which means he could believe James product is for instance 0 (3*0*0*0) or 4 (4*1*1*1) so therefore Jack is not sure what James has. Jack can figure out that James cannot have for instance 1 or 2 or 9*9*9*9 or any combination of digits that will let Jack know all digits. (I believe that for each number with range=3 there's more than one possible combination for John, I haven't checked this detail yet)
So J﻿ack says: I don't know all the digits﻿﻿ but James﻿ ﻿﻿doesn't either.﻿

James has 4 (product) so he could guess the number is 2211 or 4111. James can figure out that John’s sum is 6=2+2+1+1 or 7=4+1+1+1. He also see for instance that 6+0+0+0=6 and 7+0+0+0=7 so James, from his point of view,  knows that John can’t figure out the four digits from John’s sum (6 or 7).
So James says: I don't know all the digits ﻿but John doesn't either.

John has 7 (sum) so he may guess that possible numbers are 7000, 6100, 5200, 5110, 4300, 4210, 4111, 3220 or 3211.
John, with his sum=7 can figure out that Jack’s range is 7,6,5,4,3 or 2 from the possible numbers above. John can also see that James product is 0, 4 or 6.
John then sees that if James product is 0 then James, from his point if view above, would have to see the possibility that John could have had sum=0 and the number would then have to be 0000. So John decides that James product is not 0 since James said "John doesn't know".
John then sees that if James product is 6 then the number would have to be 3211. Then Jack’s range would be 2. But then, from Jack’s point of view, Jack should have seen that Jack’s range=2 means that James product could be 2 and James, from Jacks initial point of view, should have been able to see that 2111 is the number. But Jack said: "James﻿ ﻿﻿doesn't know".
John then sees that if James product is 4 there is one matching number, 4111, in the list.

So John draws the conclusion that James have product = 4 and Jack has range=3 and the number is 4111.

4-1=3 (Jack)
4*1*1*1=4 (James)
4+1+1+1=7 (John)

I haven’t yet figured out the very last part, if Jack must be able to tell the number when John says that he has the solution.

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59 minutes ago, Ghideon said:

My guess is:

Reveal hidden contents

4111

So you assume that Jack knows that the range is 3.

How does Jack think about the possibility that James knows that the product $$a\cdot b\cdot c \cdot d$$ is equal to $$250$$?

Edited by taeto

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35 minutes ago, taeto said:

So you assume that Jack knows that the range is 3.

Yes.

35 minutes ago, taeto said:

How does Jack think about the possibility that James knows that the product a*b*c*d is equal to 250?

Thanks! +1
I missed that possibility because of lack of attention to details. Therefore the solution is incorrect.

(I'm beginning to believe my english skills are too limited for these kind of riddles)

Edited by Ghideon
edit: x-posted, deleted statement that became incorrect

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The "range" has to mean something different from $$r=a-d.$$ If the value of $$r$$ is all that Jack knows, then his statement that James cannot know the answer leads to elimination of all the possibilities $$r=0,1,\ldots,8$$ in turn: If the solution happens to be $$a=r+1$$ and $$b=c=d=1,$$ then James can actually figure it out just from knowing the product $$p=r+1.$$ That leaves only the possibility $$r=9,$$ so $$a=9$$ and $$d=0.$$ Then James cannot be right in stating that John will not be able to know the answer, e.g. if the sum is $$s=27.$$ (Another way to argue is to consider that James has no more information than John already knows, so it cannot be true that John can find the solution after hearing what James has to say.)

Hence "range" should mean the same as the pair $$R:=(a,d);$$ the hyphen between biggest and smallest in the problem description is a hyphen, not a minus.

Edited by taeto

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On 2/14/2019 at 5:05 AM, Umurbaba said:

Jack only know the range of the digits(biggest -smallest).

Can you clarify if that is a hyphen or a subtraction: in other words does this mean the range is the pair of digits ("biggest to smallest") or the difference between them ("biggest minus smallest")?

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And is it possible to post an exact version of the question, as a .jpeg image or similar?

Usually this style of puzzle is standard and has step-by-step arguments and solutions posted online. This particular one is interesting in its combination of number theory and logic. And I have not found it anywhere online after some googling attempts. I would be quite interested to use a simplified version in a test for students of a number theory course, or maybe a course on abstract algebra. Preferably not in an exam, since even if simplified it may still feel too difficult.

18 hours ago, studiot said:

Going back to Jack and the range.

So I do not know all the digits implies he knows at least one of them.

So when Jack says:" I dont know all the digits﻿﻿ but James﻿ ﻿﻿doesn't either." And if Jack is actually implying that he knows at least one of them.

Then what does "James doesn't either" mean exactly?

Edited by taeto

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5 hours ago, taeto said:

The "range" has to mean something different from r=ad. If the value of r is all that Jack knows, then his statement that James cannot know the answer leads to elimination of all the possibilities r=0,1,,8 in turn: If the solution happens to be a=r+1 and b=c=d=1, then James can actually figure it out just from knowing the product p=r+1. That leaves only the possibility r=9, so a=9 and d=0. Then James cannot be right in stating that John will not be able to know the answer, e.g. if the sum is s=27. (Another way to argue is to consider that James has no more information than John already knows, so it cannot be true that John can find the solution after hearing what James has to say.)

Hence "range" should mean the same as the pair R:=(a,d); the hyphen between biggest and smallest in the problem description is a hyphen, not a minus.

Surely not.

Posting the range in the format   $\left( {\alpha ,\beta } \right)$ woudl simply tell James the first and last digit, would it not?

I have seen something similar before, I can't remember if this was one of Martin Gardiner's questions.

Martin was a great author.

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Yes, exactly, the first and last digit would be the information that Jack has. Which he does not communicate to anyone else.

But if the range were to mean their difference, then Jack effectively tells everyone that the first digit is 9 and the last is 0 anyway. This way there is less information conveyed.

Not to be overly pedantic, but for those who want to google, the name is correctly spelled Martin Gardner.

Edited by taeto

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9 and 0 was what we all think it was but the answer is quite interestingly 8221

I have the question as a pdf file but sadly it is in Turkish. But translate is  accurate as me and my English teacher have checked it

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1 hour ago, Umurbaba said:

I have the question as a pdf file but sadly it is in Turkish. But translate is  accurate as me and my English teacher have checked it

Can you confirm that "range" means 7 in this case (not, 8,1)?

Can you say whether "I don't know all the digits" means that why don't know any digits, or they know some digits (but not all)?

1 hour ago, Umurbaba said:

9 and 0 was what we all think it was but the answer is quite interestingly 8221

I can't yet see any way of concluding that from the information given. (But maybe one of the smarter guys who have been thinking about this for longer, will do!)

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1 hour ago, Umurbaba said:

9 and 0 was what we all think it was but the answer is quite interestingly 8221

I have the question as a pdf file but sadly it is in Turkish. But translate is  accurate as me and my English teacher have checked it

Well I await with interest your chain of reasoning to prove that 8221 is the only answer possible.

8221 has a sum of 13, a product of 32 and what I interpret as the range of 7

some other possibilities with a sum of 13 are

4333 product = 108 and range 1

9211 product 18 range 8

So why did John choose 8221?

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Yes, 8221 is a very interesting answer. It does not lie in any of the extremes, as when having a 9 at the start or a 0 at the end, or mostly the same digits, etc.

The problem is to figure out with reverse engineering which question this might be the correct answer for.

I will attempt by pure improvisation from my own pet theory, that "range" means the pair of the smallest and the largest digits.

In this case the range would be $$R = (8,1).$$ So Jack knows this.

The information from Jack that James will not be able to deduce the answer from knowing the product $$p=32$$ does not help James or John a lot, unless there are lots of ranges for which the solution is obvious once you know the product. It is basically only ranges like $$(1,1),(5,5),\ldots,(9,9)$$ and a few more for which this is possible.

Knowing that the product is $$p=2^5$$ however will help James along quite a bit. It leaves only $$8411$$, $$8221$$,$$4421$$, and $$4222$$.

James now knows that John will be looking at a sum $$s \in \{10,11,13,14\}.$$ Indeed, John is looking at $$s=13.$$ Now 9400 is one of the possibilities that John has to think about. It is consistent with Jack saying that he does not know all digits by knowing that the range is $$R=(9,0).$$ And also with Jack saying that James will not be able to figure out the answer even knowing that the product is $$p=0.$$ The only thing that 9400 is not consistent with is when John says that he knows that Jack also can deduce the answer. To Jack, the two possible solutions 9400 and 9220 would look exactly the same, judging from the discussion between James and John. This excludes 9400 as solution.

Edit: that was garbage, please ignore. The real question is whether John can eliminate 9400 as a solution, and right now I do not see how he can.

There is a list of additional possible solutions with sum $$s=13$$ for John to consider, 9310, 9220, 8500, 8410, 8311, 8221, 7600 etc. Surely each, except 8221, can be excluded on similar grounds, no?

25 minutes ago, studiot said:

8221 has a sum of 13, a product of 32 and what I interpret as the range of 7

You really have to give up on your idea of "range".

If Jack only knows that the difference $$a-d$$ is equal to $$7,$$ he most definitely cannot ascertain that James will not figure out the precise solution once James knows the product.

Edited by taeto

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But we don't know that the sum is 13 until John has declared.

We have been asked to work it out without that knowledge.

Edited by studiot

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You are missing my point.

Let us just assume that the 4-digit number is 8221.

We agree that Jack does not know that this is the solution, but somehow he knows that the difference between highest and lowest digit is $$r=7$$, right?

We agree that the rule is that Jack does not tell James and John that the value of $$r$$ is 7, but he does tell them that he can inform them that no matter what the product $$p$$ is, James will not be able to deduce the solution, even if James knows the value of $$p$$?

James now knows that $$r$$ is not $$0,$$ because if $$r=0,$$ then James will be able to deduce from $$p=1$$ or $$p=6561$$ what the exact solution is, namely 1111 or 9999, respectively, do we agree?

James also knows that $$r$$ is not $$1$$, because if $$r=1,$$ then James will deduce the solution from $$p=2,$$ namely 2111.

And so on, until $$r=7,$$ which it would be for 8221. But again, Jack cannot say that James will not deduce the solution from knowing the value of $$p.$$ Since if $$p$$ is equal to $$8,$$ then James knows immediately that the answer is 8111.

Please tell me what you are missing here, or what it is that seems wrong to you.

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12 minutes ago, taeto said:

but he does tell them that he can inform them

Jack does not say that John can't deduce the solution, only James.

Note also at the end that no one says Jack would be able to deduce the solution.

14 minutes ago, taeto said:

James now knows that r is not 0, because if r=0, then James will be able to deduce from p=1 or p=6561 what the exact solution is, namely 1111 or 9999, respectively, do we agree?

Why can p not equal 0 ?

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18 minutes ago, studiot said:

Jack does not say that John can't deduce the solution, only James.

That is correct: "no matter what the product $$p$$ is, James will not be able to deduce the solution". Nothing about John there.

18 minutes ago, studiot said:

Why can p not equal 0 ?

If 8221 is the assumed solution, then $$p$$ is not $$0$$.

I gather that we do not agree that the rule is that Jack does not tell James and John that the value of $$r$$ is 7, but he does tell them that he can inform them that no matter what the product $$p$$ is, James will not be able to deduce the solution, even if James knows the value of $$p$$?

23 minutes ago, studiot said:

Note also at the end that no one says Jack would be able to deduce the solution

The last statement of the problem description is "John says: I just found it and Jack should be just found it as well.﻿"

It does not contradict what you are saying?

Edited by taeto

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1 hour ago, taeto said:

If 8221 is the assumed solution, then p is not 0 .

I gather that we do not agree that the rule is that Jack does not tell James and John that the value of r is 7, but he does tell them that he can inform them that no matter what the product p is, James will not be able to deduce the solution, even if James knows the value of p ?

But I don't see any justification for anyone to assume a solution of 8221 at that stage of the development.

It should be possible stae what is known (or can be deduced) from each stage as the development proceeds.

Like this

Jack starts

He knows that

He has a value for the range.

James has a value for the product

John has a value for the sum.

Nothing else since neither James nor John have yet spoken.

He deduces that

He can not solve the problem

That he knows some of the digits

That even after he has told James that he knows somee, but not all of the digits James cannot solve it with James' additional knowledge.

Thank you for making me do this, it has answered my question

Since Jack knows some of the digits, r cannot be zero as this means all the digits are the same, but does not determine which digit they all are.

Following and continuing this process it should be possible to show that 8221 is the required solution (or not) without guessing it.

That is arriving at it at the end of the argument.

Edited by studiot

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It appears that the original text was formulated in Turkish language.

Depending on the quality of the translation, it may not be safe to rely on any assumptions about the meaning of "range" or whether Jack indicates that he knows at least one of the digits.

The phrase "Jack should be just found it" could make one suspicious.

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17 hours ago, studiot said:

8221 has a sum of 13, a product of 32 and what I interpret as the range of 7

some other possibilities with a sum of 13 are

4333 product = 108 and range 1

9211 product 18 range 8

So why did John choose 8221?

I will try to take a stab.

If Jack knows only $$r=1,$$ then Jack would not have said that James cannot know the solution, because the solution might have been 2111. This rules out 4333 for both James and John.

If Jack even knows $$R=(4,3),$$  then Jack knows that the product will be one of 108, 144, and 192. Apart from 4333, another possible solution could then be 9431, 9441, or 8432, respectively, so far as James knows.

Now there is a complication: if the actual solution were 9431, then Jack would have $$R= (9,1)$$ and maybe Jack would have to admit that James might possibly deduce the solution from knowing $$p.$$ The solution could be 9751 in that case. Then 7533 is the only other alternative that has the same product. Remarkably, 7533 can be ruled out: if Jack knows $$R=(7,3),$$ then James would know the precise solution in the case 7773, which is the only 4-digit number with product $$3\cdot 7^3.$$ The same argument rules out all other solutions with $$R=(7,3)$$ and $$R=(9,1)$$ as well.

So 9431 and 9441 get eliminated. Possibilities such as 9322, 9422 and 8432  still remain though.

So far John seems unable to eliminate 4333 as a possibility. But we have not yet considered James's statement. The product 108 cannot occur in many ways: 9322, 9431 are the only two alternatives. As above, 9431 can be dismissed just from the info that Jack gave. James observes that the possible sums are 13, 16. Neither of these allows John to determine a solution, certainly this is true of the sum 13, which is shared by 8221, the (supposed) actual solution. And sum 16 has enough candidates that no exact identification is possible, it seems.

End of analysis of the case 4333. I do not see how John can rule it out as the solution if Jack knows $$R.$$ But it is tricky, and I can definitely have overlooked something.

Did I make a mistake?

Edit: I do not see how John can rule out 4333. But I do see now how both James and John can rule out 8221, as follows.

Jack knows only $$R=(8,1).$$

The solution could be 8771, so far as Jack knows. The only other number with the same product is 7742.

We eliminate 7742 by observing that if Jack knows only $$R=(7,2),$$ then James would know the exact solution if it is 7772.

Since 7742 is not a possible solution, and 8771 is the only other 4-digit number with product 392, also 8771 cannot be the solution.

We deduce that there is no solution with $$R=(8,1).$$

Edited by taeto

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9 hours ago, taeto said:

The phrase "Jack should be just found it" could make one suspicious.

I agree. Your statement made me look at the beginning of the thread again. We didn't get all rules initially, a followup said:

On 2/15/2019 at 5:22 AM, Umurbaba said:

So there is no complete English translation given yet?
Speculation: What happens if the correction only applies to Jack? So that the following initial statement

Quote

Jack only know the range of the digits(biggest -smallest).John only knows a+b+c+d.James knows﻿ a*b*c*d.

Actually was supposed to be something like: "Jack only know the range of the digits; meaning he knows that a>=b>=c>=d and he knows a - d (a minus d). John only knows a+b+c+d. James knows﻿ a*b*c*d."
In other words, is it possible that John and James do not know that
a>=b>=c>=d?
I haven't yet checked what difference this would make when trying to solve the problem.

A maybe even more speculative question:

On 2/14/2019 at 5:05 AM, Umurbaba said:

Jack, John and James tries

But then they all act in another order:
Jack says... James says... John says...

Is that change of order the names intentional or was the names mixed up in translation? I haven't yet checked what difference this would make.
Can we see the original question? (Even if it's in Turkish it might help at this point)

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