# Symmetry Inversion Riddle

Just a simple riddle that occurred to me the other day:

What sees itself upside-down in a mirror, but not back-to-front?

Edit

The answer is with respect to the object itself, and only requires one mirror. It is important that it doesn't see itself back-to-front in the mirror: that is there is no back-to-front reflections (as far as it is concerned) as well as its upside down one.

• Rather than making the assumption, would the mirror you're referring to be a generic, flat mirror? – Brent Hackers Dec 21 '16 at 7:38
• this riddle is evil, I can't think of any witty explanation of how you can have a reflection without it being back-to-front..... – Spacemonkey Dec 21 '16 at 17:57
• @BrentHackers Yes, their is nothing special about the mirror. – BM- Dec 21 '16 at 22:27
• @Spacemonkey, nor really the reflection. The riddle is about what sees its reflection upside-down... – BM- Dec 21 '16 at 22:28
• @BM- , absolutely; but the problem is that any reflection is back-to-front, it's essentially what makes it a reflection. – Spacemonkey Dec 22 '16 at 15:28

My thoughts:

! An exclamation mark!

Because:

The 'i' in 'a mirror' is an exclamation mark upside down! Also, there is no exclamation mark in 'a mirror', so it is not back-to-front (disclosing the 'o')!

• That's a really cute answer! I love it! Unfortunately, I didn't tag the riddle as word-play, so it's not what I have in mind. – BM- Dec 20 '16 at 23:50
• @BM- I'll keep it here for future reference anyway, unless it gets heavily DV'ed – boboquack Dec 20 '16 at 23:51

Anything which is

upside down (therefore sees itself as upside down when reflected)

but also

has left-to-right symmetry (therefore doesn't see itself inverted left-to-right).

For example:

• Not quite. The first statement is definitely part of it (although the thing I am thinking of can look at a mirror rightside-up and still see itself upside down) – BM- Dec 20 '16 at 23:12
• The second statement, whilst is usually true for the thing I'm thinking of, is actually not necessary. However, you are also missing the principal attribute that allows something to satisfy the riddle. – BM- Dec 20 '16 at 23:14
• @BM- You might need to edit the question then, in order to ensure there's only one possible correct answer. Riddles with multiple equally valid answers are likely to be closed. See also the tag wiki for riddles. – Rand al'Thor Dec 20 '16 at 23:18
• ai'thor I don't think it's too broad under the specifications of riddles. What I am thinking of exhibits a particular attribute that allows it to satisfy this riddle. It's possible that there are other things that also have this attribute; however, I have been unable to think of anything else that does! – BM- Dec 20 '16 at 23:48

A (film) camera.

Why:

The image is inverted on the film but is not reversed when viewed from the correct side of the film.

• This isn't what I'm looking for, but as an answer to the puzzle, it is very fitting! Technically my answer fits for even in front of the lens, but I like this one. – BM- Dec 21 '16 at 22:32
• But I suppose, as @BrentHackers pointed out, anything with a lens technically works. Which would therefore include humans, and make the puzzle trivial... – BM- Dec 21 '16 at 22:34

my guess is

a mirror placed directly above your head, if you see up you'll see yourself upside down and there is no back-to-front reflections.

• This is along the lines of the three mirrors in a corner. Your reflection would be upside down, but to see it like that, you'd have to look up at the mirror. Your reflection of your face would still be back-to-front – BM- Dec 21 '16 at 22:30

It could be

a flounder

By putting ourselves in his position,

Looking into the mirror, you could say that he thinks he's flipped vertically instead of horizontally.

Clarification:

As commented by OP: "If I were him, and facing a mirror straight on, I'd see myself back-to-front: my left-side's reflection is my right-side." If I were floating vertically (my right-side is up, left-side is down toward any assumed floor) then I'd know my left side is down, but my reflection has his right side down, therefore - my reflection is me, but upside down.

• I like this answer as well. I woke up thinking about something with eyes only on one side of their face. – Jim Dec 21 '16 at 16:23
• This is very close. Certainly the right way of thinking, but could you please clarify his point-of-view? If I were him, and facing a mirror straight on, I'd see myself back-to-front: my left-side's reflection is my right-side. If I saw myself side on, I'd see again an identical me, with my left-flank as its right-flank, and vice versa. To me, that's still back-to-front, not upside down. I can't see how having eyes like that changes that point-of-view. – BM- Dec 21 '16 at 22:46
• And I am aware that my comment up there could be misleading, Certainly the thing I am thinking of would see itself the same way, but there is a reason as to why it interprets itself upside down. I am struggling to think why this thing can make the same rationalisation. – BM- Dec 21 '16 at 22:47

An ambigram

For example...

In the mirror it looks like..

Which appears to be only upside down.

• I actually already had that thought, and tried it out. the ambigram can most definitely see itself in a back-to-front orientation if it so chooses. – Rubio Dec 23 '16 at 5:45

An additional answer, which depends not on the what, but on the how:

In a room where there is a mirror on a wall and a ceiling, if you look at your ceiling-reflection in the wall mirror, you will see yourself upside down and not L/R inverted.
OP says this isn't the goal, so:

Updated

In a room where there is a mirror on a two adjoining walls and the ceiling, if you face directly at one of the wall mirrors and then look at your other-wall-reflection in the ceiling mirror, you will see yourself upside down and not L/R inverted.
If your shirt says "boo", you will see it say "poo".

• Technically, in this situation, you also see yourself back-to-front, albeit with a different reflection... – BM- Dec 20 '16 at 23:55
• I'm not sure I agree. If I wear a shirt that on its front reads "boo", just looking in a mirror I will see myself right-side-up but my shirt says "ood" — it is back-to-front. In my scenario, the reflection I see is vertically inverted - the head is the bottom of the image - and the shirt says "ooq", which is what you'd see if I was upside down. Or are you insisting the shirt should read "poo" ? – Rubio Dec 21 '16 at 0:12
• no, I'm saying in this situation, you will see three reflections, one of which is upside down. However, the other two, especially the one on the vertical plane, will be back-to-front. – BM- Dec 21 '16 at 0:20
• I asked a straight question; I think a straight answer is warranted. For a person wearing a shirt which says "boo" on it to satisfy the constraints of your riddle, what must it appear to read in the mirror? And you say there would be three reflections, which is true; are you requiring one and only one reflection which reads in that indicated fashion to satisfy the riddle? – Rubio Dec 21 '16 at 0:26
• Yes. Your inverted bit does satisfy the first part of the riddle, in which the object would see itself upside down in the reflection-of-the-reflection. However, the riddle specifies that they don't see themselves back-to-front in the mirror, which would contain both reflections. It is true, though, that you could arrange the two mirrors to satisfy the riddle by ensuring that vertical mirror is situated too high to see your direct reflection. – BM- Dec 21 '16 at 0:39

My answer complete with photo proof:

My answer is 2 over 5, written in a digital font. It appears as 5 over 2 in its reflection.

• Your artwork is lovely. Such subtleties of purple and blue! :) (But this is qualitatively no different from a 3 or E - it can read its reflection as if it were upside down, but could just as well choose to read it as if merely back-to-front. OP says, "The answer, though, would not have that choice, it would interpret its reflection as upside down." Since a mirror by its very nature causes back-to-front flipping, I eagerly await knowing what thing seeing itself in a mirror cannot choose to see what is of necessity actually there.) – Rubio Dec 21 '16 at 6:42
• 3 or E. Either one in the mirror appears as itself either rotated 180 degrees or as itself back-to-front, same as your fraction does. – Rubio Dec 21 '16 at 7:05
• If you put 3 AND E in the mirror together stacked on top of one another (like my fraction and my 2 and 5 year olds artwork :)) then I'd concede these were the same (along with i and !) because there is no left to right inversion from this perspective, just the appearance that the 2/5 3/E or !/i have traded places. – Jim Dec 21 '16 at 7:24
• Sure. But it's still a matter of perspective. Since both left/right and upside-down have identical appearances, you could argue successfully for either interpretation. OP says that for the intended answer, this cannot be so argued. – Rubio Dec 21 '16 at 7:30

This could be

an upside-down cake. A round cake has no back-to-front reflections. If it's the normal way up then it will see an upside-down cake in the mirror; if it's the other way round then it will see an upside-down upside-down cake. Either way, it's upside-down...

• and I do love upside-down cake... – BM- Dec 21 '16 at 22:48

A single coloured, plain spherical ball.

Reason being.

No matter which way the ball is angled, it looks the same. Be it upside down, horizontal, it looks the same, thus it sees its self upside down, whilst also arguing it hasn't reflected at all due to the fact that the ball is (in my assumption a perfect sphere)well... the same all around.

• The problem with this answer is that the argument to "it's upside-down", and "it's not back-to-front" require conflicting rationalisations. It's premise for arguing one way could not allow it to argue the other, without Orwellian DoubleThink. – BM- Dec 21 '16 at 22:59

Could it be

This guy

Or this

line on my very vague graph?

Or

everything, because

Otherwise all I can think of is

something like an egg timer or something where it's state changes based on its orientation, and is therefore always upside down?

Seems like a good puzzle to me...

this guy? although technically his head would be the right way up and actually, he'd still kind of see himself flipped horizontally...

• I had forgotten about lenses, which technically makes this puzzle trivial... Perhaps the word 'sees' is a little too broad then. The riddle is referring to being able to interpret your reflection at point of observation, rather than the literal method of image capture! Still, excellent answer. Additionally, while I like the state-change answer, the answer's interpretation is constant. – BM- Dec 21 '16 at 22:56

OP commented:
This puzzle was brought upon from the philosophical question that comes from that observation [that every reflection is back-to-front inverted]. Why? Why is a reflection back-to-front; why does a mirror favour flipping across the vertical axis and not any other?

The truth is, of course, that it doesn't favor anything.
When we look at an object as it faces toward us, left from its perspective is on our right.
When we look at its reflection as it faces a mirror, left from its perspective is now on our left.
The object's top/bottom/left/right, as seen: $$\small\begin{array}{ccc}\bf{by\ someone\ else}&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &\bf{in\ a\ mirror}\\\begin{array}{rcl}&top&\\right&&\ left\ \\&bottom\end{array}&&\begin{array}{rcl}&top&\\\ left\ &&right\\&bottom\end{array}\end{array}$$

If that object is turned upside down, here are its own top/bottom/left/right, as seen:

$$\small\begin{array}{ccc}\bf{by\ someone\ else}&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &\bf{in\ a\ mirror}\\\begin{array}{rcl}&bottom&\\\ left\ &&right\\&top\end{array}&&\begin{array}{rcl}&bottom&\\right&&\ left\ \\&top\end{array}\end{array}$$

What's on our right side as we look in a mirror will be on the right side of the reflection we see in that mirror. But when we look at an object, its left side is to our right, so if it is facing away from us and toward the mirror, what we see left-to-right in the reflection is the exact left-to-right reverse of what we would see if it were facing toward us, the way we normally see it. We're used to thinking of a horizontally arranged set of symbols as having a "front" and a "back" when reading, as that's how we read it, so L/R inversion can be thought of, and called, back-to-front, but that's potentially misleading; what we really mean is left-to-right inverted.

Here's where things get interesting. When you look in a single, regular mirror, you always see yourself left-to-right inverted from how other people see you, or how a picture of you taken by a camera would see you. OP is asking what object would see that left-to-right inverted image and interpret it as an upside-down (top-to-bottom inverted) image with no left-to-right inversion.

I would have expected the answer thus to be an object that, looking at its reflection, sees $\small\begin{array}{rcl}&bottom&\\\ left\ &&right\\&top\end{array}$
That is, it sees what it would see if it could see itself face to face, but the other one of itself was turned 180° to be upside down. That is, in actual fact, precisely what you would see if you were to hold a mirror above your head, parallel with the ground and facing down, and then look up at it.

Someone actually gave that answer and OP replied, "[...] Your reflection would be upside down, but to see it like that, you'd have to look up at the mirror. Your reflection of your face would still be back-to-front." So even though your actual left side is on the reflection's left side, while your bottom is on the reflection's top side, this is not a solution, per OP.

However, there's perhaps some mischief at work here. For a human to see their reflection they do indeed have to look up at the mirror, so they see their own face in the reflection pretty much the same as if they just looked at a mirror. That is, it's basically upside-up and left/right inverted.

Now, imagine something that can see itself in a mirror placed above it, without having to turn its face to look at the mirror. Something that has a distinct front and back.

Something like a crocodile.

The crocodile has amazing field of vision. It can basically see anything in front, behind, or above it; indeed, just about anywhere except directly in front of its snout, or directly behind it.
The crocodile has a "front" (its snout) and a "back" (its tail).
It has a "top" (its back) and a "bottom" (its belly).
And if it were to look at itself in a mirror above itself, pointing downward, what would it see?
The reflection would have the "top" closest to itself and the "bottom" farthest away — in other words, the reflection is upside down.
The reflection's "front" would still be ahead of the crocodile's eyes, and the "back" still behind them — in other words, the reflection is not back-to-front.
The specific wording of the riddle is suggestive, in that it never actually says left-to-right inverted, only front-to-back. As noted earlier, we're accustomed to thinking of left-to-right as being front-to-back, but that needn't actually be the case. However, even if you insist on left/right inversion not being observed by the crocodile, I point out again that seeing an exact image of oneself upside-down places one's left side on the left side of the image; this, too, is the case for the crocodile seeing itself in a mirror above itself.

If this isn't a solution, I dare say that with no other activity on this question but mine in the past week, PSE is largely stumped and it may be time for OP to demonstrate a valid solution exists. Cuz if this isn't it, I'm calling shenanigans.

• I did mean left-right reversal as back-to-front. It is a good rationalisation though. – BM- Jan 3 '17 at 10:32

I think you're a

Spoon.

Why?

You can see yourself back and forth and it has a part that mirrors you upside down.

• Well, from a certain point of view. I suppose I did say that perspective was important. Unfortunately, said object also sees itself back to front; like Reno's answer, this one relies on multiple reflections/mirrors. The answer only requires one mirror - one reflection – BM- Dec 21 '16 at 5:51

So this puzzle has remained unanswered for more than a couple of weeks, and interest seems to be dying down. At Rubio's request, I'll demonstrate that this puzzle does indeed have an answer.

The major hint I've provided within the comments is that the object in question cannot interpret its reflection to be back to front, only upside down. I also made a reference to another philosophical conundrum, why do we see ourselves back to front in a mirror? What is so special about reflection about the z-axis?

I am referring to an object's local Cartesian Coordinate system:

• X-axis: extends out left and right
• Y-axis: extends front and back
• Z-axis: extends up and down.

The answer to this is that there is nothing special about the z-axis at all as far as the mirror is concerned. In fact, if you think about what a reflection is there is no reversal around any axis: light that comes from my right is reflected back to me on my right; light from my left to my left; light from above me above me; light from below, below.

So why do I interpret that as back-to-front?
Answer: because I spatially rotate the image and compare it with what I expect to see.

Say I have a doppelganger standing next to the mirror, facing the same direction I am, and it turns to face me so I can compare it with my reflection. It naturally will rotate around the z-axis, and so the comparison will show that while the top and the bottom are in the correct positions, the left and the right are now opposite.

This rotation around the z-axis is so natural to us, we hardly see it as an option. We naturally exist in a world with gravity, and whilst we may have 6 degrees of freedom (translation and rotation of the three local axes), we predominately move along the y- and around the z-. To turn and face someone is to rotate around our z-axis. And therefore, the comparison we can draw from comparing our reflection to that of someone facing us, is to see it back-to-front.

Now the Riddle itself asks, is there anything (common) in our world for which this isn't true? Is there something that cannot rotate around its z-axis, but perhaps can rotate around another?

The answer is a Foosball figurine.

The reason is:

A foosball figurine is constrained in its reference frame from rotating around the z-axis, but is often facing either direction. It rotates around its x-axis. Suppose again a particular figurine was facing a mirror and its doppelganger (maybe in the middle row). Its doppelganger now turns to face it, by rotating around the only axis it can, the x-axis. The comparison now shows that the top and bottom have swapped places, but the left and right have not. Thus, the figurine sees itself upside-down but not back-to-front!

Many of the answers provided looked at changing the placement of the mirror. In particular, in placing the mirror above you and looking up. This is interesting, because if we examine this situation in the same way as before, our "doppelganger" is now rotating around the y-axis (our front still faces forward, from our perspective). Our comparison would show that our reflection has swapped both up-and-down and left-and-right!

• Shenanigans! No, no, and no! First of all, I assume we're considering the "particular figurine" and its perspective; what its middle-row doppelganger does is irrelevant. Pretty much every foosball table figure is top/bottom asymmetrical (and many are actual "men"), so there is a definite top and bottom (and often "left" and "right"); if it looks (past its doppelganger) at the mirror, it definitely sees itself upside up, and left/right reversed. I don't understand the handwaving you're doing by rotating the doppelganger, but "our" reflection is unchanged and does NOT have the asserted property. – Rubio Jan 3 '17 at 11:12
• That's the point. No where in the puzzle description did I say that our perspective was relevant. It isn't. The figurine's is. It most certainly does have horizontal asymmetry, and indeed our interpretation of its reflection is back-to-front, because our understanding of a reflection is rotating about the z-axis. The figurine cannot do this. It cannot experience seeing itself face-to-face with the top up. All experiences with seeing its comrade's faces will be inverted. Therefore, it's interpretation of its reflection must be upside-down. – BM- Jan 3 '17 at 11:21
• I said "our" meaning the figurine's (I put us in its place with its perspective). You're claiming that its only experience seeing another of its kind is to see it inverted as that's the only orientation one could be in and be faced its way; poppycock. Besides the obvious counterexample of the opposing figurines, which do point its way and are right side up, you ignore that, even though it only sees its doppelgangers' faces when they are inverted, it still sees them right side up from behind when they face the same direction. It knows that orientation is not only possible, but frequent. Foul :) – Rubio Jan 3 '17 at 11:31

It is

3 or E
These are top/bottom symmetric such that left/right inversion is the same as 180° rotation.
(Indeed, anything for which that property is true sees itself upside-down.)

• However, something with that property also sees itself back-to-front when it sees itself upside down. – BM- Dec 20 '16 at 23:51
• Not also - it can either see itself back-to-front, or upside-down, but not both together—making it a matter of perspective. But see my other answer, where something can indeed see itself strictly upside-down only. – Rubio Dec 20 '16 at 23:53
• Perspective is very important for the answer: an 'E' looking at itself in a mirror has a choice of interpreting the reflection as upside down or back to front. The answer, though, would not have that choice, it would interpret its reflection as upside down. Something else with a different perspective, however, may disagree with its interpretation... – BM- Dec 21 '16 at 0:02