Question

Could someone please help me with this question? http://i.imgur.com/L72z7VF.jpg

Answer 1

what I would do to approach this question is to try to eliminate answer dwarves that are obviously not right.

Answer 2

we know that grumpy's statement is true... but we can easily eliminate another dwarf.
let's look at doc's guess

Answer 3

if a number is a multiple of 30, it must be a multiple of the factors of thirty. both 10 and 15 (sleep and happy's guesses) are factors of thirty, however

Answer 4

can you tell me what this means @Sepeario

Answer 5

@Sepeario , I would appreciate a reply of some sort to indicate that you are following my thoughts.

Answer 6

this means that there are three dwarves that are correct. @inkyvoyd

Answer 7

sorry about the late response

Answer 8

really? grumpy said that only 2 dwarves can be correct...

Answer 9

that means that if Doc's guess is correct, than there must be 3 dwarves that have guessed correctly.
The only problem is that Grumpy mentioned that only 2 dwarves guessed correctly.
In essence, this is a contradiction, so there is no possible way that doc's guess is correct

Answer 10

yes. that was what I was trying to say @inkyvoyd

Answer 11

so according to our elimination, which remaining dwarves might be right?

Answer 12

sleepy, happy, dopey and sneezy

Answer 13

there are a few ways to go about doing it from here. I did it with prime factorization:
multiples remaining:
10,12,15,18
10=2*5
12=2*2*3
15=3*5
18=2*3*3
To give a rigorous argument, I believe we can eliminate another dwarf simply by analyzing him.
Let's say 10 is a correct guess. However, 10 must be paired with another guess, because there are two correct guesses.
if 10 is paired with 15, then the number must be a multiple of 2*3*5=30... which is impossible, because 30 is another guess. What about if 10 is paired with 12?

Answer 14

I think maybe that could be it?

Answer 15

@inkyvoyd

Answer 16

really?

Answer 17

if 10 is paired with 12, then what must the number be a multiple of?

Answer 18

60 @inkyvoyd , which is also a multiple of 30

Answer 19

So then shouldn't it be 12 and 18?

Answer 20

that's what I got, after applying similar reasoning to 15.

Answer 21

Ok, thanks very much @inkyvoyd

Answer 22

np :).
just don't ask anyone else to solve the questions :P

Answer 23

http://www.maths.uq.edu.au/~uqowarna/papers/Junior_2010.pdf