# Horizontal and vertical

Let’s say you are blindfolded and taken into a gravity free spherical room where you lose all sense of direction and then you are presented with two square shaped sheets of metal. In addition, it’s promised that if a person standing outside the spherical room looks at the sheets in their present positions, then he will say that one of them is horizontal (meaning it’s lying down) and the other is vertical (meaning it’s standing up on one of its edges). Next, you are given the permission to take those sheets in your hands and inspect them in whatever way you want. Your task is to somehow decide which one was horizontal and which one was vertical when they were given to you. To help you out, you have been connected on phone to a person who is standing outside the sphere and who can see the two sheets through a small whole. You are allowed to ask him as many questions as you want, but the only problem is that he speaks French. So if you point to a sheet and ask him if it is vertical or horizontal, he will tell you the right answer, but you won’t understand it. Can you still accomplish your task? (I am assuming you do not understand French.)

I just came up with this puzzle a few minutes ago. I want to know if the answer I have in mind is trivial or not.

## 12 thoughts on “Horizontal and vertical”

1. Ayvee says:

OK, couple of questions:

1. Is the sphere completely opaque from the inside? And is the hole through which the French dude can see you one-way?

2. Can the French guy understand you? If yes, is there any reason why the following exchange is inadmissible?

FG: Non.
You: Is the sheet I’m holding in my hands right now horizontal?
FG: Non/oui (as the case might be)
You: *if answer was no, stop, else repeat with other sheet*

3. Does the way the question is stated matter? (i.e. is it something like http://xkcd.com/169/)

2. Vinayak says:

I have done quite a horrible job in describing the puzzle precisely. So let me try to make it unambiguous.

You are taken into a spherical room that’s gravity free and opaque except for one tiny hole in it, which you can’t spot because it’s really tiny, but there is a guy standing outside looking through it. The guy can see through it because he is standing really close to it. When you enter the room, you see that there are two metal discs floating still. (I am intentionally changing it to discs, because otherwise there was an answer that I had not intended.) You are promised that one of them is exactly horizontal and one is exactly vertical. Your task is to figure out which one is what.

The problem with the series of questions you have stated is that you do not understand French. So you don’t know what Non means and what Oui means.

The French guy can understand you. The way the question is stated might matter, but I did not intend it to be. In fact, to make things simpler, let’s forget the French guy. Let’s just say that the sphere is completely opaque, with no hole and you are not connected on phone to anyone. The only assistance you have is that whenever you point to a disc with your index finger, a voice in the sphere tells you whether it’s horizontal or vertical, but it says that in some alien language which you don’t understand.

3. Vinayak says:

By the way, you have to figure out what the orientations of the two discs were when you first saw them. You are allowed to lift them, move them, rotate them, kick them, eat them, whatever you want, in the end you have to find which one was horizontal and which one was vertical in the beginning.

4. Ayvee says:

Re the non and oui: my question was, does the French guy or the voice always use the same word for yes or no, and can you tell if he/it is giving you the same response for two different queries?

5. Vinayak says:

Yes, for both.

6. Ayvee says:

Well, then something like what I said above should work, right?

You: *some tautology, for e.g.:* If a statement is true, is the statement true?
Voice:
You: *hold one disc* Is the disc I’m holding right now horizontal?
Voice: or

Depending on whether the answer you just received was the same as or was different from the answer to the first question, you can tell if it was a yes or a no.

7. Vinayak says:

Yes, damn. I should fix the communication protocols between you and the voice. Let’s say that all you are allowed to do is to point at the discs and whenever you do that, the voice says “Horizontal” or “vertical” in its own language depending on if the disc you are pointing at is horizontal or vertical.

8. Ayvee says:

OK, the voice tells you whether the /current/ orientation of the disc is horizontal or vertical, correct?

Rotate each disc 90 degrees from its original position around the two canonical axes parallel to its surface, and ask the voice to announce the orientation in each of these two positions (not very clear, but I’m guessing you know what I mean). The voice will give you the same answer to the two questions for the horizontal disc, and different answers for the vertical.

9. Vinayak says:

Yes, the solution I had in mind is similar to this, except, I do not understand what ‘canonical’ means here. There are many axes parallel to the surface of the discs.

But anyway, the key idea is that it seems that horizontal and vertical are /really/ different. For example, a cube has only two horizontal faces but four vertical faces.

10. Ayvee says:

Canonical: drop perpendiculars to the surfaces of the two discs. That gives you two of the axes, and you can get the third using a cross-product (right hand/left hand rule).

And anyway, as for whether this solution is obvious: it sort of is once you figure out it’s a math problem. Q: What is the only thing that differentiates horizontal and vertical? A: “Vertical” is one pair of directions, “horizontal” is infinitely many.

But I thought it was a /physics/ problem at first — how do you get the effects of gravity to leak into the room? Maybe a high speed rotation or something? But no, you said “gravity free”, not just spaceship in equatorial orbit or whatever, so it had to be something else.

So, yeah, maybe some more misdirection would make this more difficult.

11. Ayvee says:

By the way, you said there was another (unintended) solution with square sheets — what was that?

12. Vinayak says:

Yes, that’s exactly what I had in mind. I was pretty surprised when I discovered that the obvious difference was not the only difference between horizontal and vertical.

A friend of mine said something that sort of makes sense – horizontal and vertical are actually two dimensional concepts and these weird things are happening because we incorrectly use it describe the orientation of three dimensional objects. In two dimensions, things are simple. There are two directions, one horizontal and the other vertical. In three dimensions there are three directions. So the correct way of importing this thing to three dimensions would be to have three things – horizontal, vertical and something else. It’s not very precise, but it makes some sense.

The unintended solution with the square sheet was this – since I had defined ‘vertical’ as standing straight on one of its edges, you could just rotate each sheet along an axis perpendicular to the surface. The vertical one would no longer be standing straight on one of its edges, but the horizontal one would still be horizontal.