# Can You Find the Number?

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There is a 4 digit number which spells like "abcd". Jack, John and James tries to find this number. Jack only know the range of the digits(biggest -smallest).John only knows a+b+c+d.James knows﻿ a*b*c*d.

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

James says: I dont know all the digits ﻿but John doesnt either.

John says: I just found it and Jack should be just found it as well.﻿

What is the four digits ?

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Question: is the problem to find the set $$\{a,b,c,d\}$$ of digits, or the actual 4-tuple $$(a,b,c,d)$$ (i.e. the 4-digit number "abcd")?

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Sorry my bad. a>=b>=c>=d. You should find the 4 digit number

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I'm sure there must be more to it than this, which might be clearer if it was written in proper English,

but this line of reasoning fits the given information.

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

John says: I just found it and Jack should be just found it as well.﻿

John knows the sum and the only sum that he would be able to declare I know the answer is if the answer is a = b = c = 0 so that the sum = 0.

This is the only way the sum of zero can be achieved.

Once John has said he knows, Jack knows this as well as the product (which must equal zero)

But before he knew what John knows ie they are all the same, he could not know how many zero's there are making the product zero.

Jack of course knows that all the digits are the same because he knows the range is zero but does not know which same digit it is.

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Here is the real challenge. After John says I am done, Jack should find the number. Also as James said John didnt know the number. But John came to this conclusion by Jack and James's comments.

Also there is only one true answer. If the sum is 1 then the olny number is 1000

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5 hours ago, Umurbaba said:

Here is the real challenge. After John says I am done, Jack should find the number. Also as James said John didnt know the number. But John came to this conclusion by Jack and James's comments.

Also there is only one true answer. If the sum is 1 then the olny number is 1000

I think my answer fits all the statements in the order given.

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19 hours ago, Umurbaba said:

After John says I am done, Jack should find the number

Also there is only one true answer. If the sum is 1 then the olny number is 1000

So John works out the answer, knowing that neither James nor Jack  have enough information but knowing that the sum is 1.

John can only do this if there is a unique combination that can only occur one way and can match his sum to that number.

But since we are not told this sum and there are four possible unique sums that can only occur one way and also fit the information given to all.

a>=b>=c>=d.

That is

a) 1000
b) 0000.
c) 9998
d) 9999

Clearly if John knows that the sum = 1 he declares (a) as the only correct answer.
But he would also solve it if he knew that the sum was zero and declare the correct answer to be (b)

Or if he knew the digit sum to be 35 or 36 he could declare (c) or (d).

He would not be able to solve this if his sum was  greater than 1 and less than 35 since there are multiple ways to meet the abcd restriction for these sums

But we do not know the sum so must look at his statement that James should also be able to solve it now.

If James knows that the product is zero which means that either (a) or (b) will suit, but he cannot distinguish

So the answer cannot be 1000 or 0000

or that the product is 5832 or 6561 when he can declare (c) or (d)

So we must look again at what Jack says.

Now Jack says I don't know all the digits but I know that the greatest minus the least is 1, which gives answer (c) or 0, which gives answer (d)

I cannot see a way for us to distinguish between answers (c) and (d), all though each of Jack, John and James have enough to do this at the appropriate time.

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By the way this was from a math exam. In couple of days they should declare the correct answer.They are still saying there is one unique answer no matter how many times I asked them

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As always with these the devil is in the detail

Please post the exact question, exactly as written.

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Yes please. I am confused about the use of the term "range". Conventionally it would mean the set of numbers $$\{a,b,c,d\},$$ but it looks like it is used for the difference $$a-d$$? It could also mean the interval $$[d;a]$$ of numbers from smallest to largest?!

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Just now, taeto said:

Yes please. I am confused about the use of the term "range". Conventionally it would mean the set of numbers {a,b,c,d}, but it looks like it is used for the difference ad ? It could also mean the interval [d;a] of numbers from smallest to largest?!

Given the greater than or equal to chain of relations, this means that 'a' must be the largest and 'd' the smallest, although the may actually be equal.

That is how John can distinguish between 1000 and 0100 and 0010 and 0001.

Only the first is allowable under this condition.

a>=b>=c>=d.

That is the assumption I am working on anyway.

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The condition $$a\geq b\geq c\geq d$$ eventually got made explicit by the OP, hence it should be safe to assume.

Which assumption do you work with for the use of "range" though?

Maybe the question is on a higher level, where you have to figure out the unique use of "range" which produces a single possible answer. That would be nasty.

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

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

The range of the digits, not the range of the numbers abcd is specified.

The largest possible digit is 9, the smallest is 0 so the range is either 0,1,2,3,4,5,6,7,8,or 9

The largest digit of these must appear in position a and the least in position d.

I see no other possible interpretation of that part of the information, unless the question was incorrectly reproduced.

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If "range" is simply $$r:=a-d,$$ then we have:

Lemma. $$r \in \{7,8,9\}.$$

Proof. Jack knows that the product $$p:=a\cdot b\cdot c \cdot d$$ is not enough information to determine $$abcd$$. But if the range is one of the numbers $$1,2,4,6$$, then the value $$(a,b,c,d) = (r+1,1,1,1)$$ would be the unique quadruple with $$p = r+1,$$ since $$r+1$$ is a prime, which means that the value of $$p$$ determines the solution uniquely. Also $$r=0$$ is excluded, because of $$(9,9,9,9)$$. For $$r=3$$ the quadruple $$(5,5,5,2)$$, and for $$r=5$$ the quadruple $$(7,7,7,2)$$, are determined by $$p$$.  That leaves the three possible values stated.

Does that make sense?

Edit: $$r=7$$ is excluded, due to $$(9,9,9,2 )$$ as well.

Edit: after a further think, $$r=8$$ is also excluded, because of $$(8,0,0,0)$$. Because now James and John both deduce $$r \geq 8$$ from the information that Jack is giving. John's sum $$s = a+b+c+d \geq a + 0 + 0 + 0 \geq r \geq 8$$ can be equal to 8 only in this particular case. Which means that James cannot deduce that John will not be able to get the correct answer. The upshot is that we have $$r=9$$ and $$p = 0$$, which implies $$a=9$$ and $$d=0.$$

Edited by taeto

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But now all quadruples $$(9,9,9,0), (9,9,8,0), (9,1,0,0), (9,0,0,0)$$ seem possible?! They are the possible $$(9,b,c,0)$$ for which the sum $$9+b+c+0$$ with $$9 \geq b \geq c \geq 0$$ determines $$b$$ and $$c$$ uniquely. I must be doing something wrong ￼

Edited by taeto

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24 minutes ago, taeto said:

But now all quadruples (9,9,9,0), (9,9,8,0), (9,1,0,0), (9,0,0,0) seem possible?! They are the possible (9,b,c,0) for which the sum 9+b+c+0 with 9 \geq b \geq c \geq 0 determines b and c uniquely. I must be doing something wrong ￼

It is interesting following you looking at this problem through the other end of telescope from me.

You are starting with Jack and the range.

I am working backwards from John and the sum.

But John must know that {1111} is not an option since James would have already found that since it is the only {abcd} that yields a product of 1.

So I am concentrating on which sum they all know can only be formed in one way.

I still think these are 0, 1, 35 and 36.

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2 minutes ago, studiot said:

So I am concentrating on which sum they all know can only be formed in one way.

I still think these are 0, 1, 35 and 36.

I am not sure how you mean that.

The sum 0 cannot be in the range of solutions, because James says (truthfully, I hope) that John cannot yet know the solution. And when (0,0,0,0) is still in the range of possible solutions, so far as James can ascertain, it would be a lie if James says that John cannot possibly know the solution yet at that point. At least James does not have enough information to make that statement.

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10 minutes ago, taeto said:

The s﻿um 0 cannot be in the range of solutions, be﻿cause Jame﻿s says﻿ (truthfully, I hope) that John ca﻿nnot yet know the sol﻿utio﻿n﻿.

How does that follow inevitably?

I prefer to use list of potential solutions rather than range to avoid confusion.

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1 minute ago, studiot said:

How does that follow inevitably?

I prefer to use list of potential solutions rather than range to avoid confusion.

Okay, I thought that this is how the range of solutions would be interpretated anyway.

The sum 0 being the solution means that $$(0,0,0,0)$$ is the solution. But James says that John will not be able to figure out the solution. However, John knows that the sum is 0. So James is lying when he says that.

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35 minutes ago, taeto said:

Okay, I thought that this is how the range of solutions would be interpretated anyway.

The sum 0 being the solution means that (0,0,0,0) is the solution. But James says that John will not be able to figure out the solution. However, John knows that the sum is 0. So James is lying when he says that.

Yes I agree.

That would mean that the product is also zero.

But James only knows the product and that any {abcd} with a zero in any position will yield a zero product.
And there are many possibilities for this.
So if James' product is zero he cannot know which {abcd} produced it.

Further he also knows that the products  are not {9999} = 6561 and {9998} = 5832

or he would have reported solving the problem since those are the only way to obtain those particular numbers.

So he says

1) he does not know the answer.

2) He also says that John does not know the answer.

When he says (1) he is not then lying about (2) because  two of the proposed solutions John is considering, {0000} and {1000} ,have zero products.
But they have different sums.

So John can then ascertain whether his sum is 0 or 1, thus leading to unique solutions {0000} or {1000}

What I can't see is how that allows him to say that James can now solve it because he did.

I am assuming he does not pass on the sum or solution information.

Edited by studiot

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I am not sure where you are going. Certainly neither 0000 nor 1000 can be solutions, based on the fact that John would immediately realize this, based only on the value of the sum.

If any of the guys are lying, there is no way the problem makes sense, so I will assume they are all truthful.

Conversely, do you see how any of the numbers 9990, 9980, 9100, 9000 would not qualify? I see a problem exactly where you identify it as well, namely that neither of these purported solutions would allow John to deduce that Jack will also know it by then. Each of them looks the same from Jack's point of view, does it not?

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50 minutes ago, taeto said:

I am not sure where you are going. Certainly neither 0000 nor 1000 can be solutions, based on the fact that John would immediately realize this, based only on the value of the sum.

Why wouldn't they?

Until John has spoken, no one knows what he knows or doesn't know.

And John speaks last. And he immediately solves the problem.

This is why I say something is missing in the presentation of the question.

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8 minutes ago, studiot said:

Until John has spoken, no one knows what he knows or doesn't know.

James may actually know. I suspect you are missing this point.

John has to work with the information available to him. James knows how much information is available to John (except for the precise value of the sum). So James may well be entitled to make statements about whether John can or cannot know the answer, based on having the sum available.

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4 minutes ago, taeto said:

James may actually know. I suspect you are missing this point.

John has to work with the information available to him. James knows how much information is available to John (except for the precise value of the sum). So James may well be entitled to make statements about whether John can or cannot know the answer, based on having the sum available.

For each and every value of digit sum, S : such that  $2 \le S \le 34$ I can find at least 2 different {abcd} that achieves this sum.

How does John distinguish between these in each case since he cannot find out from the sum alone ?

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30 minutes ago, studiot said:

I can find at least 2 different {abcd} that achieves this sum.

Great. So we can take $$S=2$$ as an example sum.

The possibilities are obviously just 1100 and 2000.

At this point Jack knows that the range is 1 or 2.

If the range is 1, then Jack will have to admit that 2111 may be a solution, in which case James may know the true answer for certain since the product is $$p=2.$$ Jack has no basis for stating that James cannot know the answer.

If the range is 2, it is similar, since 3111 may be the solution.

In either case, knowing that the range is 1, or knowing the the range is 2, does not provide Jack with enough information to ensure that James cannot figure out the answer just from knowing the product $$p$$.

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