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"Look at these flags and tell me-" Said Grandpa flashing a paper with many Flags on it.

"Which one does not belong to the group? All, except one, have one interesting commonality. One property they all show except one," He continued.

"And before you tell me the obvious square one, let me be clear. It has nothing to do with the shape or size of the flag itself, whether it is a country flag or not, the colors or where the countries come from-- the geography I mean. And no obvious things like only Japan has a circle or Israel has a star. We all know each flag is a different design."

"That makes it hard to guess," I complained.

"Well. Here is a hint. The commonality kind of relates to a 3 digit number-- chew on that one" He said with a smile.

I am more confused. Which one is it and why?

enter image description here

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    $\begingroup$ I'm sorry but I find the question misleading. You explicitly say it has nothing to do with shape, but the accepted answer is about rotational symmetry $\endgroup$
    – lhk
    Jun 16 '20 at 10:14
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    $\begingroup$ @lhk I think OP meant "shape or size of the flag", but now that you mention it, it indeed may be worth clarifying. $\endgroup$
    – oAlt
    Jun 16 '20 at 12:46
  • $\begingroup$ @ihk: I edited the clarification in. In future best to go ahead edit it yourself, in cases like this where it greatly changes the interpretation. $\endgroup$
    – smci
    Jun 17 '20 at 0:08
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The odd flag out is

The EU Flag

Because

It is the only flag which does not exhibit Order-2 rotational symmetry - that is, if you rotate the other flags halfway round to turn them on their heads, then they will look the same, but the stars on the EU flag would be upside down!

Which means the 3 digit number is

180°, the angle through which they can be rotated

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    $\begingroup$ Oh - I was slightly wrong; it doesn't remain the same if flipped horizontally. $\endgroup$ Jun 15 '20 at 14:58
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    $\begingroup$ Are you sure? Gur Fg. Cngevpxf synt (erq K) vf abg pragrerq va gur Fg. Naqerjf synt (juvgr K). Syvccvat vf abg flzzrgevpny. $\endgroup$ Jun 15 '20 at 22:58
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    $\begingroup$ @majorbanzai - Gur nafjre vf ebgngvba, abg syvccvat. $\endgroup$
    – Sphinxxx
    Jun 16 '20 at 0:20
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    $\begingroup$ I understand, but Grandpa needs to acknowledge that this puzzle has more than one answer. $\endgroup$ Jun 16 '20 at 16:05
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    $\begingroup$ @Steve All the shapes are quadrilaterals where all interior angles are 90° - as such, this does not make any distinction based on shape, nor size. $\endgroup$ Jun 24 '20 at 14:12
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The odd-flag-out is

the UK

Because

It cannot be flipped horizontally, nor vertically, and still look the same. The EU flag cannot be flipped vertically, but it can be flipped horizontally. All other flags would look the same flipped in either direction.

The three-digit number is

180 - representing the degrees of flipping

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    $\begingroup$ I like this answer except (Rot13: lbh qba'g ersre gb fbzrguvat gung unf orra zveeberq nf orvat syvccrq ol bar uhaqerq naq rvtugl qrterrf fb gur yvax gb n guerr qvtvg ahzore vf rkgerzryl grahbhf (VZB)...) $\endgroup$
    – Chris
    Jun 18 '20 at 12:09
  • $\begingroup$ Better explanation of this answer (ebg13: Vg'f gur bayl bar gung ybbxf qvssrerag jura ebgngrq bar uhaqerq naq rvtugl qrterrf nobhg gur synt cbyr) $\endgroup$
    – Steve
    Jun 26 '20 at 6:35
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The Japanese flag is the only one that your grandpa drew with a frame (of color #000).

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    $\begingroup$ Great observation. Grandpa has a different answer though $\endgroup$
    – DrD
    Jun 15 '20 at 14:17
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Considering the accepted answer, I would respectfully suggest it is either "wrong" or at least not unique, because there are at least 3 valid answers all involving the same basic operation:

  • Only one flag looks different when rotated 180° in the plane (EU: @Chronocidal answer)

  • Only one flag looks different when rotated 180° about the flagpole (UK: @majorbanzai answer)

  • Only one flag looks the same when rotated 270° (or 090°) in the plane (Swiss - this answer).

It is unclear how (or if) 'Grandpa' intended one of these to be "more unique than the others".

Original phrasing of this answer:

All but one look different when rotated by 270° (or indeed 90°, but that's only a 2 digit number!)

The odd flag out from THAT definition is also the most different in other respects:

It is the only flag labelled in the question by an adjective rather than a noun.
As acknowledged in the question, it is the only square flag (implied by the rotation by 270°, but a non-symmetrical square flag also exists - Grandpa could have included the Vatican City flag to force this answer over the other one, but not sure how he could have disambiguated the other way)

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  • $\begingroup$ Surely this is discounted by the condition "It has nothing to do with the shape or size of the flag itself"? $\endgroup$ Jun 24 '20 at 13:22
  • $\begingroup$ @Chronocidal when simplified to a single statement, the accepted answer and this one differ only in the 3-digit number mentioned when defining the operation that separates one flag from all the others. $\endgroup$
    – Steve
    Jun 24 '20 at 14:04
  • $\begingroup$ Your answer hinges entirely on the fact that one of the flags is a square shape, and the others are not. Otherwise, the UK and Japanese flags could fulfil the condition about as well as the Swiss Flag (which sometimes is flown as a rectangle) $\endgroup$ Jun 24 '20 at 14:10
  • $\begingroup$ @Chronocidal only to the same extent that your answer hinges entirely on the rectangular nature of the other flags - same basic operation, two different 3 digit numbers => two different answers. $\endgroup$
    – Steve
    Jun 24 '20 at 14:16
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They can all be rotated 180 and look identical, except the EU flag

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The correct answer is the EU flag, but the reason in the accepted answer is wrong.

The reason has nothing to do with the design of the flags. All the flags except for the EU belong to countries which have 3-digit numeric country codes defined in ISO 3166 (see https://www.iban.com/country-codes for example).

The EU is not a country, and therefore does not have a country code.

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    $\begingroup$ Unfortunately the question excludes this answer in its setup - it doesn't depend on "whether it is a country flag or not"... Otherwise this would be valid! $\endgroup$
    – Stiv
    Jun 17 '20 at 18:01

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