I Don't Know Your Number. Wait, Now I Do!

Logic Level 2

Alice thinks of two positive integers, secretly tells Bob one, and secretly tells Charlie the other. Then the following conversation occurs:

Alice: "The product of my numbers was 8. Or maybe it was 16."
Bob (to Charlie): "I don't know your number."
Charlie (to Bob): "I don't know your number."
Bob (to Charlie): "I don't know your number."
Charlie (to Bob): "I don't know your number."
Bob (to Charlie): "I know your number!"

What was Charlie's number?

16 2 8 4 1

2 solutions

Shaun Leong
Jan 26, 2016

Let b be Bob's number and c be Charlie's number.

Firstly, $b \in {1,2,4,8,16}$ and $c \in {1,2,4,8,16}$ .

From statement 1, b cannot be 16 since then Bob would know c=1.

$b \in {1,2,4,8}$ and $c \in {1,2,4,8,16}$

From statement 2, c cannot be 1 or 16 since then Charlie would know for sure b=8 or 1.

$b \in {1,2,4,8}$ and $c \in {2,4,8}$

From statement 3, b cannot be 1 or 8 since then Bob would know for sure c=8 or 1.

$b \in {2,4}$ and $c \in {2,4,8}$

From statement 4, c cannot be 2 or 8 since then Bob would know for sure b=4 or 2.

$b \in {2,4}$ and $c \in {4}$

$c=\boxed{4}$

(Thanks to @Luke Johnson-Davies for spotting additional numbers to rule out in my original solution)

A missing detail here is that after statement 2 you can also eliminate c = 16 for the same reason you eliminated b = 16 for the previous statement (Charlie would know b = 1). From this, you can also eliminate b = 1 after statement 3 and then c = 8 after statement 4, leaving c = 4 as the only possible value (and also with this, after statement 5 you leave $(b,c) = (2,4),(4,4)$ which is inconclusive in terms of determining b, so b is not necessarily 4).

Luke Johnson-Davies - 5 years, 4 months ago

I can't believe I missed that out! Thanks for pointing that out - the solution has been edited.

Shaun Leong - 5 years, 4 months ago

In the sentence about the statement 3, you probably meant that "for sure c=8 or 2" and not "for sure c=8 or 1"

Юрій Ярош - 11 months, 2 weeks ago

Aareyan Manzoor
Jan 26, 2016

if Bobs number was number 16, Charlie's number would have to be 1, so since Bob doesn't know what Charlie's number is, Bob's number is in the set (8,4,2,1).
If Charlie's number was 1, Bobs number must be 8 or 16(already eliminated).For same reason as Bob, his number isn't 16 either. So, charlies set is (8,4,2).
If Bobs number is 1 then charlies must be 8, hence rejected. If his number was 8 charlies must be 2, hence rejected as well. His new set is (4,2).
If charlies number is 8, Bobs must be 2. If it is 2 his must be 4. so his new set is (4)
Charlie finds the number! $\boxed{4}$ .

The only mistake is that in 4. , if Charlie numbers is 2, Bob's must be 4 and not 8.

A Former Brilliant Member - 5 years, 4 months ago

Thanks i have edited my solution

Aareyan Manzoor - 5 years, 4 months ago

I tried writing down method like yours , but its a bit brute forced because i cant memorise how every event linked with each other.

The bracket with numbers indicate the solving sequence (1) , (2) , (3) etc.....

B = 16 C =1----------------------------(1)	B = 8 C = 1----------------------------(2)
Bob : i know your number	Bob : I do not know your number
	Char: I know your number

B = 8 C = 2----------------------------(3)	B = 1 C = 16----------------------------(4)
Bob : I do not know your number	Bob : I do not know your number
Char: I do not know your number	Char: I know your number
Bob : I know your number

B = 1 C = 8---------(5)	B = 2 C = 8----(6)(refer 5)	B = 4 C = 2----(7)(refer 3)
Bob : I do not know your number	Bob : I do not know your number	Bob : I do not know your number
Char: I do not know your number	Char: I do not know your number	Char: I do not know your number
Bob : I know your number	Bob : I do not know your number	Bob : I do not know your number
	Char: I know your number	Char: I know your number

(refer 4) C must be 8 or Conversation 4 is repeated

(refer 5) IF B = 1 when C = 8 , the conversation will end up as Number 5 , since it didnt , B must be 2.

(refer 3) IF B = 8 when C = 2 , the conversation will end up as Number 3 , since it didnt , B must be 4.

B = 2 C = 4-------------------Answer/Remainder	B = 4 C = 4-------------------Answer/Remainder
Bob : I do not know your number	Bob : I do not know your number
Char: I do not know your number	Char: I do not know your number
Bob : I do not know your number	Bob : I do not know your number
Char: I do not know your number	Char: I do not know your number
Bob : I know your number	Bob : I know your number

The only possible/remainder conversation left would be this , therefore Charlie has number 4 , otherwise the conversation will follow the pattern from 1 - 7.

Ne-ko Nya - 5 years, 4 months ago

I Don't Know Your Number. Wait, Now I Do!

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