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Here's a quickie:

I own a shirt that has a logo on the chest on my left side. This morning I accidentally put it on inside out and back to front. Where did the logo go?

Please explain your answer as clearly as possible.

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Inside out and back to front puts the logo

behind your back, on the left, against your skin.

because

the front of the shirt is against your back, facing forwards, so just imagine the front of the shirt behind you without any rotation.

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  • $\begingroup$ With the front of the shirt against your back that's the only possilbe arrangement $\endgroup$ – Jasen Jul 21 at 1:27
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I tried it for real and:

Left-hand side, back, inside.

Imagine a hollow cylinder with a front and back and a logo at the front left. To turn the cylinder inside-out, make a vertical cut through one edge of the cylinder, and flip the disconnected edges to make a new connection the other way around, which involves inverting the curve of the cylinder components. This has the effect of reversing the orientation of the cylinder, so now the logo is on the front right and inside. 'Back-to-front' is a $180^\circ$ turn of the cylinder, which places the logo back inside left.

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  • $\begingroup$ I attempted to try it for real but I got all tangled up and ended up wearing the shirt upside down. - Where was the logo? $\endgroup$ – A. I. Breveleri Jul 21 at 3:10
  • $\begingroup$ Bottom right inside back. But how did you get your arms in? $\endgroup$ – JMP Jul 21 at 4:08
  • $\begingroup$ No pic, didn't happen :D (jk, upvoted for actually trying it) $\endgroup$ – mu 無 Jul 21 at 15:16
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There are three variables, with two possibilities for each:

  • Left (as intended) or right
  • Front (as intended) or back
  • Outside (as intended) or inside

The shirt starts with the logo outside in front at the left. Any operation flips the values of two of these variables -- flipping just one would entail transporting the shirt into a higher dimension and back again with its sense reversed. You say the shirt is inside-out, so the logo is on the inside; and back to front, so it's at the back. Those two variables are not as intended, so they got flipped, so the remaining variable is

OK: the logo is on the left, as intended.

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  • $\begingroup$ I like your explanation very much. You could also add the third axis (top-bottom) to the system, because the action of turning the shirt inside out is more easily understood along that axis. I wrote up an answer to explain how I visualise this process. $\endgroup$ – Jaap Scherphuis Jul 21 at 14:10
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Turning something inside out implies you flip it vertically (but inside of course), which places the logo on the right chest but inside if the T-shirt is otherwise worn right. If it's also worn front-to-back, it's flipped horizontally on top of that, so the logo will be placed on the left side of your back but inside.

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I don't know if this is what you mean by "as clearly as possible", but it's easy to solve this problem using group theory. What we're dealing with is essentially "the group of the rectangle", also known as $V_4$ or $C_2\times C_2$ or the Klein 4-group or various other names.

Let's divide your shirt into four sections: front outer (1), front inner (2), back outer (3), back inner (4).

  • Back to front swaps 1 with 3 and swaps 2 with 4.
  • Inside out (assuming you still wear the front to the front and the back to the back) swaps 1 with 2 and swaps 3 with 4.

So the combination of both

swaps 1 with 4 and swaps 2 with 3. We're looking at the permutation representation of the relevant group.

That means your logo, originally on the outside of the front of your shirt, is now

on the inside of the back of your shirt, and therefore still touching the left-hand side of your body (namely, on the right when looking outwards from your back).

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While several answers have used a kind of short-cut reasoning, explaining directly what the final state has to be, I like the challenge of visualising all the steps, going from wearing it normally to wearing it inside out and back to front. I think the easiest steps to visualise are as follows:

1) You are wearing the shirt normally. The logo is on the left side of the chest, on the outside of course.
2) Someone else grabs hold of the shirt at the bottom hem, and pulls it upwards over your head, turning it inside out in the process. The shirt is held there above you, inside out and upside down. Clearly the front it still at the front, the left is still at the left. The logo moved straight upwards, is still at the front left, but is now on the inside of course.
3) Now the shirt does a front flip or back flip. This turns it the right way up again, but such that it becomes back to front as required. The logo stays on the left during this flip, so it ends up on the back left, still on the inside of the shirt.

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