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Write a sentence that has the following properties:

  1. Is a palindrome
  2. Is true
  3. Can be used as a template to generate an infinite number of sentences which are both palindromes and true

As a bonus, write a sentence in a language other than English that still has the above three properties.

(There is a solution for both challenges. Hope you'll have fun finding it!)

EDIT: congrats to @noneuclideanisms for finding the answer for English. Who can now find the sentence in another language? Hint: it's a loose translation of the English answer.

EDIT 2: Since nobody managed to find the solution in another language, I've added it below as an answer.

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  • $\begingroup$ Can you explain point 3 about the template? $\endgroup$
    – Techidiot
    Feb 14, 2017 at 18:43
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    $\begingroup$ The sentence can be easily modified to generate an infinite number of sentences that are also palindromes. These sentences can be of arbitrary length. I can't be more precise without giving a hint (which I'll do if someone asks). $\endgroup$
    – dr_
    Feb 14, 2017 at 18:47
  • $\begingroup$ For property 3, I see two possible interpretations/rules. Rule 1: A B B A can be extended to A B A B B A B A by nesting it in itself. This can be done an infinite number of times. Rule 2: A B B A can also be A C C A or A B C B A, by altering the B B section, you can create a number of sentences. $\endgroup$
    – tfitzger
    Feb 14, 2017 at 21:20
  • $\begingroup$ Writing this as a comment because I don't think it can really be classed as true. "A man a plan a canal panama". $\endgroup$
    – Andy
    Feb 15, 2017 at 9:46
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    $\begingroup$ @dr01 Okay, got it. Thanks for explaining $\endgroup$
    – Shokhet
    Feb 16, 2017 at 14:35

13 Answers 13

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I think the intended answer was

"x", sides reversed, is "x"

because it can be infinitely extended by doing

" 'x', sides reversed, is 'x' ", sides reversed, is " 'x', sides reversed, is 'x' "

infinitely.

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  • $\begingroup$ Yes, that was it. The way to generate infinite palindromic sentences from it is just to replace the ends e.g. "abc (...) cba". Now, can you find the answer in another language? $\endgroup$
    – dr_
    Feb 15, 2017 at 8:24
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    $\begingroup$ I don't like this because it relies on quotation. I mean, you could replace one "x" with "asdfg" and the other with "gfdsa" and it would work equally well. That feels kinda cheaty. $\endgroup$
    – Gareth McCaughan ♦
    Feb 15, 2017 at 14:53
  • $\begingroup$ @GarethMcCaughan how could you use asdfg and gfdsa when they're not palindromic? surely the left and right hand sides of the sentence must be equal, regardless of reading direction? $\endgroup$
    – Matthew Layton
    Feb 15, 2017 at 15:03
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    $\begingroup$ @series0ne Nope. "Sides reversed", it says. $\endgroup$
    – Gareth McCaughan ♦
    Feb 15, 2017 at 15:05
  • $\begingroup$ Well, just put a palindrome in for x and it doesn't feel as cheaty. Like racecar! $\endgroup$
    – Bailey M
    Feb 16, 2017 at 6:20
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How about

A Toyota's a Toyota.

Because

It's a palindrome. It's true (a tautology, even). It's a template for an infinite number of sentences like so:
A Toyota's a Toyota's a Toyota's a Toyota['s ...]

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    $\begingroup$ That's from the good old days when Gertrude Stein won the Open Toyota Palindrome Contest. $\endgroup$
    – M Oehm
    Feb 14, 2017 at 18:59
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For the bonus:

Yo soy.

which is in

Spanish. (meaning "I am", which I think is a pretty safe assumption)

Explanation:

Yo soy.
Yo soy y yo soy. ("I am and I am.")
Yo soy y yo soy y yo soy.
et cetera

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    $\begingroup$ Maybe something similar words in Russian: Ya i ya i ya... (this assumes 'ya' means 'I am' in the sense of 'I exist', which I'm not sure it does...). In Cyrillic it would be a palindrome: Я и Я и Я... $\endgroup$
    – DyingIsFun
    Feb 14, 2017 at 19:28
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In italian,

Otto ama Anna e Anna ama Otto.

which means:

Otto loves Anna and Anna loves Otto.

Is true, It generates infinite sentences and it's in a foreign language.

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    $\begingroup$ Nice! +1 Now, Otto can be replaced by any palindrome Name ;) Bob,Elle, Hannah, Nolon and so on. Names could be anything and may be called infinite :) Welcome to Puzzling! $\endgroup$
    – Techidiot
    Feb 15, 2017 at 10:29
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    $\begingroup$ Are you sure that Anna loves Otto back and he's rather not been friendzoned, or similar? :) $\endgroup$
    – dr_
    Feb 15, 2017 at 11:46
  • $\begingroup$ I'm pretty sure they love each other $\endgroup$
    – Francesco Salvatori
    Feb 15, 2017 at 18:06
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    $\begingroup$ Unless you can prove that they love each other, property #2 is not satisfied. +1 nonetheless for a nice Italian palindrome. $\endgroup$
    – dr_
    Feb 16, 2017 at 9:38
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Here's one I got in Hindi(Indian)

सा रे ग म प ध नी सा नी ध प म ग रे सा ..... they are called Sargam which is equivalent to Solfège.

It is a palindrome.
It is a rhythmic sentence.
It goes to increate infinite loops.
Start from second and end at second last you get another palindrome which is a rhytmic sentence as well.
नी सा नी at the deepest makes a rhythmic sentence as well.
Infinite number of songs can be composed in any language using this rhythmic sentence.

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    $\begingroup$ Could you add an English translation please? $\endgroup$
    – Richard
    Feb 16, 2017 at 8:20
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I think the following at least qualifies for the bonus:

1 + 1 = 1 + 1

It does have all three properties you asked for, but it's debatable whether that is in English or not. (I certainly read it in English, but many others may not.)

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    $\begingroup$ Given the language tag, I wouldn't think this is acceptable. $\endgroup$
    – greenturtle3141
    Feb 14, 2017 at 21:44
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    $\begingroup$ Math is the language of the sciences.... :P $\endgroup$
    – M.Herzkamp
    Feb 16, 2017 at 15:51
  • $\begingroup$ (Spoiler Warning)      But, hey, why not say something slightly less trivial, like “1 + 0 = 0 + 1” or “1 + 11 = 11 + 1”? OK, you have to extend them chunk-wise (“1 + 0 = 0 + 1 = 1 + 0 = 0 + 1”), but that’s valid under the rules of this question. Or you could go to “1 + 111 = 111 + 1”, etc. $\endgroup$
    – Peregrine Rook
    Feb 16, 2017 at 15:52
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    $\begingroup$ Congratulations, you managed to make a true sentence not in one, but all languages! (OK, maybe not all, but certainly many). $\endgroup$
    – Florian F
    Jun 24, 2022 at 15:38
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I'm not sure whether this counts, but in the language of Boolean algebra, where $T$ represents true and $\land$ represents logical conjunction, the following is trivially a palindrome that evaluates to true:

$T$

We can use the following as the 'template':

$T \land T$

which is also a palindrome that evaluates to true. To expand, replace each $T$ with the template, and iterate. Here's the first iteration:

$T \land T \land T \land T$

As a bonus, it's not only a palindrome - its mirror reflection gives you the same statement (other than the angle of the italics slant).

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Here's the answer for a non-English language:

In Italian: "abc" a ritroso sortirà "cba"

(literally: "abc", backwards, will result in "cba").

This template, in the same way as the sentence "abc" side reversed is "cba" (@noneuclideanisms' answer) can be used to generate infinite palindromic true sentences. Just replace "abc" with any word/sentence and "cba" with the reversed word/sentence.

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My first thought stretches the definition of sentence slightly, but fits the second point to a T.

$x \Rightarrow x$ which is tautological, so is always TRUE. The argument for it being a sentence is that it reads as "X implies X." which is valid grammatically, but no longer a palindrome.

To generate infinite sentences

$f(0)=x \Rightarrow x$; $f(n+1)=f(n) \Rightarrow f(n)$

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Elu par cette crapule
==> which means in French: "elected by this scoundrel"

infinite sentence:

Elu par cette crapule Elu par cette crapule Elu par cette crapule

etc...

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I stumbled over this older post. Here is another simple one.

"Emordnilap a ton si x" is not a palindrome.

For the generic case, replace x by any palindrome, like this one for example.

And it can be translated to any language.

"Emordnilap nu sap tse'n x" n'est pas un palindrome.

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What about Turkish:

ey edip adanada pide ye

It means

Edip(Turkish Male name)! Eat pide(a kind of Turkish pastry) in Adana(a city in Turkiye)

And then infinitely

ey edip adanada pide ye,ey edip adanada pide ye, ey edip adanada pide ye..... He can eat many many pastry :)

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    $\begingroup$ I don't know much about Turkish, but in English, 'I eat pides in Adana, I eat pides in Adana, I eat pides in Adana, I eat pides in Adana.' is not a valid sentence... $\endgroup$
    – boboquack
    Feb 16, 2017 at 7:30
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    $\begingroup$ As an imperative, this sentence has no "truthiness", so requirement #2 is not fulfilled. $\endgroup$
    – M.Herzkamp
    Feb 16, 2017 at 15:54
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Not on Not on Not On Gets repetitive when extended, but ask any computer more than once and that's what you'll get.

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    $\begingroup$ It's not very grammatically correct though, is it? $\endgroup$
    – Rand al'Thor
    Feb 16, 2017 at 1:29

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