# Regain your bearings from the logicians

You have landed on an island of fun-loving logicians (hmmm ...) and don't know how to find your way home. After asking the first person you meet for directions, you are led to a secret, mystical place with a large stone engraved with the following drawing replete with directions:

"I want to go south," you explain. "Is this drawing correct?"

"Judge for yourself," answers the native. "I can only tell you that one of the arrows points south, but I cannot tell you which one. I cannot tell you how many arrows point in the right direction either, or you would know which way to go."

Show that you are bright enough to regain your bearings (whilst keeping your marbles) by working out which arrow points south.

[First fully explained answer carries the day.]

The "Pet Shop Boys" could help you too, if only you knew how.

• Does that hint completely give away which arrow points south? – Gordon K Aug 14 '15 at 23:12
• sounds like just a matter of trying N in any of the 8 arrows, then counting how many of the arrows are right for each of the 8 configs. Then break down which config has a unique number of correct arrows. I'm just too lazy to do it :P – Patrice Aug 14 '15 at 23:15

South is

to the left.

Proof: the key observation is the statement "I cannot tell you how many arrows point in the right direction either, or you would know which way to go." This tells us that the correct configuration, which say has $n$ arrows pointing in the right direction, is the only one with $n$ arrows pointing in the right direction.

If N is in the right direction, then none of the other arrows are.
If S is in the right direction, then none of the other arrows are.
If W is in the right direction, then none of the other arrows are.
If any of E, NE, SW is in the right direction, then so are the other two (and no others).
If NW is in the right direction, then none of the other arrows are.
If SE is in the right direction, then none of the other arrows are.
If none of the arrows are in the right direction, then south could be in either of the directions labelled N or NE.

So the native telling us that 0 or 1 arrows point in the right direction wouldn't enable us to know which way to go; any other number except 3 is impossible; and if 3 arrows point in the right direction, then they're E, NE, and SW and south is to the left.

QED.

• This is the best explanation of all. – Marconius Aug 15 '15 at 2:54

If NE is correct then E and SW are also correct - you need to go W

If NW correct then go SW

If S then go S

If SE then go NW

If N then go SE

If W then go E

The only unique number of correct arrows is the first option, so you need to go west (as in the title of the song by the Pet Shop Boys)

• I got it first! And you cheated by using the hint :-) – Rand al'Thor Aug 14 '15 at 23:28
• You only posted first 'cos I was typing the full explanation! – Gordon K Aug 14 '15 at 23:28
• True, but I think my explanation is now better! And I didn't use the hint, promise. – Rand al'Thor Aug 14 '15 at 23:29
• +1 Well done. The hint was fair game - next time I may not leave one unless the puzzle is more difficult. Since there was a doubt about answer timing, I have to go with the other answer as it's more complete. – Marconius Aug 15 '15 at 2:58
• @Marconius - I would have arrived at the same answer with the same strategy in the same amount of time without the hint in this case. – Gordon K Aug 15 '15 at 7:28

You need to go West. Of all the orientations of this compass, all but one has either none or one direction pointing to the right direction. The one where west is pointing down has 3 pointing in the right direction (SW, NE and E)

• You don't go West. You go South. It just looks like West. Seriously though, you're right - but I got there first, and I think my explanation is now better :-) – Rand al'Thor Aug 14 '15 at 23:29

Let's replace the symbols with numbers:

N = 1
NE = 2
E = 3
SE = 4
S = 5
SW = 6
W = 7
NW = 8

If the direction difference between two arrows is equal to the number difference, either both of them are correct or not. If we write the number difference-1 between the adjacent arrows, we can easily see if it's the case when we add up consecutive few such numbers to check whether the sum equals 0 mod 8. As we can see in this picture, only SW, NE and E (6, 2, 3) allow each other to be correct.

If one of the remaining 5 arrows is correct, then no other arrow is, so we can't find out which one points to the south even if we're told only 1 arrow is correct.

If none of them is correct, we still have 2 possibilities left, so we can't either.

Then the aforementioned trio must be right, so we must follow W (not the real west of course!).