Today is my 40th birthday and I will be celebrating by sharing a puzzle with you all. I recently stumbled upon an abstruse image, and apparently the words in main body of the image all have something in common; I think you have to figure out what exactly Π is.

Edit: Removed 'SHIP' from Π and corrected it with 'KINSHIP' (thanks to @hexomino for spotting this).

enter image description here

I have attacked the word list with the usual ciphers, but to no avail. The image was in a collection of seemingly unrelated pictures; maybe they'll be of use to you?

Image 1 enter image description here Image 2 enter image description here

Word list in text form:


N.B. It is not my birthday and I am not 40 years old, I simply added this to create a narrative. I wonder why I chose the age of 40 in particular?


The answer to the riddle below will give you a similar clue to Image 2. The background, crown, etc., are all important. enter image description here

Another subset of Π


Hint 2: This hint provides a similar answer to Hint 1 but is less cryptic.

What do the words "homo", "cell", "mega", and "micro" have in common?

  • $\begingroup$ I think Π is the set of rot13(cevzrf. Gur thl jub qvq gur nathyne pbzchgngvba qrcvpgrq vf orggre xabja sbe uvf cevzr-ahzore fvrir. Naq Π vf n ernfbanoyr yrggre sbe fhpu n frg. Naq gur dhbgngvba va vzntr 1 vf ernfbanoyr sbe fhpu n frg. Ohg) that's all I've got, and it's not enough. $\endgroup$
    – msh210
    Commented Jan 6, 2022 at 2:37
  • $\begingroup$ @msh210 Good thinking so far; you're close. Image 2 and the new hint may help you. $\endgroup$
    – user78105
    Commented Jan 6, 2022 at 5:02
  • $\begingroup$ The hint I think is "water" ... $\endgroup$
    – WhatsUp
    Commented Jan 6, 2022 at 21:11
  • $\begingroup$ Yes perhaps "United Kingdom, Malta, Bermuda, and Gibraltar" have many things in common, but the background colour and crown must be related to the answer! $\endgroup$
    – user78105
    Commented Jan 7, 2022 at 3:28
  • $\begingroup$ @msh210 The common property of the words in the set Π is that they rot13(nyy raq va C, J be Q) - Picture 1 bears this out. And yes, as you allude to, the original image depicts rot13(Rengbfgurarf' zrnfherzragf sbe pnyphyngvat gur enqvhf bs gur Rnegu). I guess the question now is why? $\endgroup$
    – Stiv
    Commented Jan 7, 2022 at 10:26

1 Answer 1


A word is in the set $\Pi$ if and only if

The number obtained from typing the word into a phone keypad is prime.
For reference, here is an image of the keypad.
enter image description here


We press each digit corresponding to the number of times needed to get that letter, so for example,
GOD = 46663
ADVANCED = 23888266222333
UNWORSHIP = 886696667777777444447
OXIDATED = 66699444328333
And all of these are prime (as are the rest of the words)

The image of Earth

As others have discovered this is a visual description of Eratosthenes' computation of the curvature of Earth. The sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit

Hint 1

We know that none of these words will be prime because they end in letters corresponding to either an even number or 5.

Hint 2

This says KEYPAD with the first image representing a "key" and the second image representing Paddington Station (whose code is PAD)

What do "homo", "cell", "mega" and "micro" have in common?

We can append the word phone to each to make a new word.

The crown picture

This one foxed me a little but I think the connection is the English language.

Edit: As OP confirmed, this actually refers to the red telephone boxes used in these places - hence the red colour (very clever and a fact I didn't know)

Astonishing coincidence

When I answered originally, I specified we get each letter by pressing the digit once. I checked a few examples to confirm this. The first I checked was GOD which gives 463, a prime.
Then, I decided to check two of the longer words as it would be unlikely that they would be prime if I were wrong. ADVANCED gives 23826233, a prime, and UNWORSHIP gives 869677447, a prime.
I should have checked some more examples before posting because apart from SOW, ZAX and A these are the only ones that yield a prime for both methods. Just surprising that these were the ones I decided to check first.

  • $\begingroup$ Excellent. Great work solving all of the hints too! rot13(Vagrerfgvat gung lbhe zrgubq jbexrq, zl zrgubq sbe pbairegvat gur jbeqf gb ahzoref jnf gb cerff gur ahzoref nf vs lbh jrer npghnyyl nggrzcgvat gb glcr gur yrggre, r.t., S -> 333 - fb TBQ jbhyq or 46663.) $\endgroup$
    – user78105
    Commented Jan 7, 2022 at 14:41
  • $\begingroup$ @draconiansomalian In all honesty, I have not checked through them all so it may not be true but I will check them now to see if this is the case. $\endgroup$
    – hexomino
    Commented Jan 7, 2022 at 14:43
  • $\begingroup$ For the crown picture rot13(Erq gryrcubar obkrf, juvpu jnf zrnag gb yrnq gb gryrcubar -> xrlcnq. Dhvgr pelcgvp bs pbhefr). Regardless of your schema I'm certain you would have made the connection anyway. Impressive stuff :-) $\endgroup$
    – user78105
    Commented Jan 7, 2022 at 14:44
  • 1
    $\begingroup$ @draconiansomalian Ah, okay, as far as I can tell all the others seem fine. Astonishingly not many work for my original answer but the ones I picked for testing all worked. I may add a comment in relation to this. $\endgroup$
    – hexomino
    Commented Jan 7, 2022 at 15:01
  • 1
    $\begingroup$ As a side note, there are actually more than 2x as many words with the property you initially proposed than there are words with the puzzle's property. rot13(~30x if ~13x, bhg bs n frg bs ~140x jbeqf) $\endgroup$
    – user78105
    Commented Jan 7, 2022 at 15:15

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