37

This is a riddle I and some friends came up with. Say I have four gnomes with different heights. They state the following:

Gnome 1: I am the shortest.

Gnome 2: I am neither the tallest nor the shortest.

Gnome 3: I am the tallest.

Gnome 4: I am not the shortest.

Exactly one of the gnomes is lying. Which gnome is the tallest?

bobble
  • 10,245
  • 4
  • 32
  • 80
user1098302
  • 617
  • 4
  • 9
  • 20
    I enjoy seeing these simpler puzzles on Puzzling because I know that I'll actually be able to do them, and not just have to read the question & answer and wonder how in the world Stiv&Co figured out the solution. – Robin Clower Oct 19 '22 at 13:51
  • 16
    ^and I mean no shade to the power users, they're brilliant and I enjoy reading their puzzles and answers, but I like being able to figure out the puzzles myself sometimes. – Robin Clower Oct 19 '22 at 13:58
  • 1
    @RobinClower don't be afraid to post your own answer if it hasn't been covered by existing answers :D – justhalf Oct 20 '22 at 09:21
  • At the circus: “Here you can see the world's largest dwarf and the world's smallest giant! They look exactly alike – two ordinary people.” – biziclop Oct 20 '22 at 20:06

3 Answers3

27

Gnome 1 cannot be lying because if 2, 3 and 4 are all telling the truth, none of them is the shortest gnome.

So gnome 1 is telling the truth, which means that gnome 4 automatically tells the truth. So the liar must be 2 or 3.

If gnome 2 is lying, he must be the tallest gnome (since gnome 1 tells the truth and is therefore the shortest). But that would mean gnome 3 is lying as well, and both can't be lying. So gnome 2 tells the truth.

All this means gnome 3 is lying (so he is not the tallest gnome). Also gnome 1 and 2 are truthfully claiming not to be the tallest gnome, making gnome 4 the tallest gnome.

Community
  • 1
Jafe
  • 78,615
  • 8
  • 162
  • 614
  • Very nicely done. I would just elaborate that final statement to perhaps complete the explanation: Gnome 4 telling the truth does not alone make Gnome 4 clearly the tallest. However, given that Gnome 3 is the one lying, we see from their own statements that Gnome 1 is the shortest and Gnome 2 and 3 are NOT the tallest, leaving only Gnome 4 to be the tallest. – Starman Oct 19 '22 at 13:53
  • A clearer way of saying "4 vf gryyvat gur gehgu"(rot13) is maybe "Vs 4 vf ylvat gura 1 naq 4 ner obgu fubegrfg"(rot13)? – Sam Dean Oct 19 '22 at 15:29
  • 2
    Hello. Newbie here. What does your comment mean @SamDean? – Ramy Oct 19 '22 at 17:28
  • 3
    @Ramy it's a cypher. Presumably since you can't use spoiler tags in comments. Checkout https://cryptii.com/pipes/rot13-decoder – THE JOATMON Oct 19 '22 at 18:52
  • 2
    @Ramy As THE JOATMON said it's a cypher that just changes letters to the ones 13 letters later. I use the following Chrome extension to easily do it. Just highlight some text, right click, select the Rot13 option https://chrome.google.com/webstore/detail/rot13/bahejodllcomogcpgnidcdaiidfmhomh – Sam Dean Oct 20 '22 at 10:17
8

Same answer, different logic.

If #1 is the tallest then he is lying. But then #3 is truthful and contradicts #1 being tallest.

If #2 is the tallest then he is lying. But then #3 is truthful and contradicts #2 being tallest.

If #3 is the tallest then he tells the truth. But then, either #1 is truthful, he is shortest and everybody tells the truth or #1 lies and nobody is shortest. In both cases a contradiction.

So, #4 is the tallest.

And it follows that #3 lies and #1 is truthful and shortest. We don't know which of #2 and #3 is taller.

Evargalo
  • 6,270
  • 1
  • 19
  • 39
Florian F
  • 29,623
  • 4
  • 63
  • 138
4

Another approach, based on figuring out which gnome is lying.

Suppose Gnome 1 is lying. Then none of the gnomes are the shortest, which is impossible.
Suppose Gnome 2 is lying. Then all of Gnomes 1, 2, and 3 are either the tallest or the shortest, which is impossible.
Suppose Gnome 4 is lying. Then both Gnomes 2 and 4 are the shortest, which is impossible.
So Gnome 3 is the liar. We therefore know that Gnomes 1, 2, and 3 are not the tallest, so Gnome 4 is the tallest.
We also know that Gnome 1 is the shortest, while Gnomes 2 and 3 are in between, in some order.

isaacg
  • 6,915
  • 1
  • 19
  • 52