# A logical question

1. Aug 12, 2008

### some_one

on a table we arrange in a line four stones.
two of them are small
two of them are large
two of them are green
two of them are purple

which one of the answers are not needed in order to determine the exact order of this stones?

1.each green stone stayes between a big stone and a small stone
2.both of the big stones are close one to the another
3.the stone on the right edge is purple
4.the green stones have the same size

i dont know where to start
i dont know how to find this answer
??

2. Aug 12, 2008

### davee123

1st, try and figure out what the order is by looking at all the clues:

1. "each green stone stayes between a big stone and a small stone"
Well, #1 tells you quite a bit. Each green stone is between two stones, one big, one small. Hence, you can conclude that:
A) neither green stone is on an end
B) both green stones are in the middle
C) because the stones are either green or purple, the stones on the ends are necessarily purple. So it goes Purple, Green, Green, Purple.
D) the 1st and 3rd cannot both match in size
E) the 2nd and 4th cannot both match in size
F) The order is necessarily one of:
- Big, Big, Small, Small
- Big, Small, Small, Big
- Small, Big, Big, Small
- Small, Small, Big, Big

That one clue told us a LOT.

2. "both of the big stones are close one to the another"
This tells us that the order is one of:
- Big, Big, Small, Small
- Small, Big, Big, Small
- Small, Small, Big, Big

3. "the stone on the right edge is purple"
This tells us nothing new. Already, we know that this is the useless piece of information. All it tells us is that the order is one of:
- Green, Green, Purple, Purple
- Green, Purple, Green, Purple
- Purple, Green, Green, Purple

But we already know the EXACT order of the colors thanks to #1. Hence, without even looking at the rest of the clues, we know the answer is #3, because #3 told us only a small chunk of what #1 already told us. So any conclusion we can make with #3 could be equally made with #1, therefore #3 is extraneous. But we can keep going anyway.

4. "the green stones have the same size"
That's helpful. It doesn't tell us anything directly about the order, but with #1 and #3, it tells us exactly what we needed to know.

All together
With #1, we know the Color order is:
Purple, Green, Green, Purple
And we know the size order is one of:

I. Big, Big, Small, Small
II. Big, Small, Small, Big
III. Small, Big, Big, Small
IV. Small, Small, Big, Big

So, after clue #1, all that remains is to figure out which order the sizes are in. With clue #2, we can eliminate possibility II, because the big stones must be close to each other. The remaining options are:
I. Big, Big, Small, Small
III. Small, Big, Big, Small
IV. Small, Small, Big, Big

With clue #4 (clue #3 tells us nothing new), we know that since the green's must have the same size, therefore the two in the middle must have the same size, hence eliminating possibilities I and IV. So the only remaining possibility is:
III. Small, Big, Big, Small

Clearly, because #3 told us something that we already learned from #1, #3 is a correct answer to the question. However, the question is (in our due diligence) whether #1 is necessary?

Suppose we didn't have clue #1. We would know:
2.both of the big stones are close one to the another
3.the stone on the right edge is purple
4.the green stones have the same size

So we'd know the size order is one of:
- Big, Big, Small, Small
- Small, Big, Big, Small
- Small, Small, Big, Big

And we'd know the color order could be:
- Green, Green, Purple, Purple
- Purple, Green, Green, Purple

But we wouldn't know which of those would be correct. Hence, #1 is clearly necessary (as it should be, since it told us a LOT of information).

DaveE

Last edited: Aug 12, 2008
3. Aug 12, 2008

### some_one

ye but we you also could say that the second answer comes from the first
because one of the conclusions that comes from the first answer is that the first and third rocks differ in size
and so does the 2nd and 4th.

in the second answer we get the same conclusion
so the 2nd one is unnecessary
where is my mistake???

4. Aug 12, 2008

### phlegmy

i havnt bothered to work it out but if you dont know where to start
to answer you question:

leave out clue no.1
can you solve with only clues 2,3,4

if not then
leave out clue no.2
can you solve with only clues 1,3,4

if not then
leave out clue no.3
can you solve with only clues 1,2,4
etc etc

eventually you'll find that you dont need one of the clues

5. Aug 12, 2008

### Werg22

Are you sure this is not a homework in axiomatics?

6. Aug 12, 2008

### davee123

Close, but not quite.

You're correct that the 1st and 3rd differ in size, and the 2nd and 4th differ in size, BUT that doesn't necessarily mean that the big stones are close together. Using only clue #1, a potential arrangement would be:

Big, Small, Small, Big

1st and 3rd differ in size (Big/Small), and 2nd and 4th differ in size (Small/Big), so it's totally valid, even though the two "Big" stones are as far apart as possible.

DaveE

7. Aug 13, 2008

### some_one

thanks
i understand your solution

8. Aug 13, 2008

### some_one

i dont know if this is axiomatics
but i need to learn solving this kind of questions
for "asat" exam in my country