# Which gnome is the tallest?

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?

• 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. Commented Oct 19, 2022 at 13:51
• ^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. Commented Oct 19, 2022 at 13:58
• @RobinClower don't be afraid to post your own answer if it hasn't been covered by existing answers :D Commented Oct 20, 2022 at 9:21
• 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.” Commented Oct 20, 2022 at 20:06

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.

• 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. Commented Oct 19, 2022 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)? Commented Oct 19, 2022 at 15:29
• Hello. Newbie here. What does your comment mean @SamDean?
– Ramy
Commented Oct 19, 2022 at 17:28
• @Ramy it's a cypher. Presumably since you can't use spoiler tags in comments. Checkout cryptii.com/pipes/rot13-decoder Commented Oct 19, 2022 at 18:52
• @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 chrome.google.com/webstore/detail/rot13/… Commented Oct 20, 2022 at 10:17

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.

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.

• I went with the same logic Commented Oct 27, 2022 at 12:38