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This is a very hot summer in Rome and you don't know what to do. You are the best detective in town but no crime seems to happen. You are all the day in front of your PC trying to solve some puzzles about parties and elephants but you can't find any solution so far. You are on the way to solving the 20th when a young policeman rushes into your room and gives you a letter.

"Get the hell out! I had the solution of that devilish puzzle and now it is gone! Get out!"

He can't understand your frustration and closes the door as fast as he could. Anyway, you opened up the letter and started to read:

Rome, 22/7/14
AHAHA Mr. Rossi, I have planted a bomb into the Colosseum! It will explode tonight at 24:00. 
I know you are a puzzle-lover, so I would give you a chance to defuse it. Go there and
and follow the instructions, every clue is important, so be focused! 
You will find a 6-fold piece of paper with some hints to crack the bomb!
Good Luck!
The evil bomber


"Accipicchia [Oh my gosh]" you said "I need to move!"

You started to run to the Colosseum and after 20 minutes you found the bomb and the instruction. There is a keyboard and a monitor linked to the bomb. On the screen, you can see a textbox which allows you to type in just 4 characters and big red text saying: "Enter the password".
You opened up the piece of paper and started to read the hint.

Let's do some math: 
2^2 = ...
3^2 = ... 
5^2 = ...
7^2 = ...
11^2 = ...
You can choose to stop here or keep on with all the prime numbers you can evaluate!
Now you should know the password!
The evil bomber

You started to think about for a while and after 20 minutes you started to type the solution that came up in your mind. After you pressed enter, on the screen you saw "The bomb has been defused!"

"Perbacco [My goodnees], I made it! Now I need to find the solution of the 20th party!".

The puzzle:
What is the password and how did he come up with that?

Edit:
Let's add some hints to help you to find the right solution.

Hint #1

The evil bomber is a PhD in Math, but he did not find any job and he started his career as a terrorist. Loving numbers, every number he gave are there for a reason, but one, just to make him struggle a little more.

Hint #2

Mr. Rossi is a fan of Math History, he loves HSM too. He found in a while which "subject" of the hints the evil bomber gave to him. It was easy, every numbers are linked univocally to that subject. It was easy to type the right answer this way.

Hint #3

During his PhD, the evil bomber studied a lot Euler, Reimann, and their Euler-Reimann Z function. He found with joy they are related to PI.

Reference

You can find at prodotto di Eulero the relation between primes and $\pi$. If you want more, you can see this one and read the demostration here


Anyway guys, I actually hope you had fun with this riddle as I had!

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  • 1
    $\begingroup$ +1 for mentioning one of the party series questions which I created. $\endgroup$
    – Michael
    Commented Nov 14, 2014 at 21:50
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    $\begingroup$ "The evil bomber is a PhD in Math, but he did not find any job and he started his career as a terrorist." He obviously wasn't paying attention when the joke was told, "What is the difference between a Math PhD and a large pizza? The large pizza can feed a family of four!" $\endgroup$
    – Michael
    Commented Nov 14, 2014 at 21:53
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    $\begingroup$ The greatest detective can think about the password in 20 minutes which the entire stack community can't and he can't even think about the password to the 20th party??? Are you kidding me?!?!?? $\endgroup$
    – Rohinb97
    Commented Nov 17, 2014 at 15:14
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    $\begingroup$ seeing so many different answers, i think safest thing is call bomb squad and let professionals to handle bomb $\endgroup$
    – user902383
    Commented Nov 18, 2014 at 9:53
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    $\begingroup$ There is a blatant flaw here which I can't believe nobody has noticed. The sum 1/2^2 + 1/3^2 + 1/5^2 + 1/7^2 + ... is not equal to pi^2/6. If you wanted to get pi^2/6 you would just take the sum of the reciprocals of the squares of all the integers, not just the primes. That is, pi^2/6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... $\endgroup$
    – Carmeister
    Commented Nov 26, 2014 at 1:30

11 Answers 11

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My guess, motivated by @nexolute 's answer:

$\pi^2/6$

(assuming the 'characters' you can input don't have to be letters or numbers). Reasoning:

As many people have already noticed, $\pi$ is important somehow. The Riemann zeta function was mentioned in the third hint, and $\zeta(2)=\pi^2/6$ is its best-known value. The number 6 is suggested by the six-fold piece of paper. And the sum of the reciprocals of all squares of primes is $\pi^2/6$.

(In the OP's original question, the numbers on the bomber's last note were (prime)^(prime) instead of (prime)^2; this is why the question generated answers such as 5^5 and took such a long time to solve. But the OP only spotted his error after the solution had already been found!)

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  • $\begingroup$ We have a winner! $\zeta(2)=\pi^2/6$ can be evalued as a sum of the reciprocal of prime numbers squared. $\endgroup$
    – Emi987
    Commented Nov 17, 2014 at 21:23
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    $\begingroup$ @Emi987 - Yes, I know, but what does that have to do with raising a prime to the power of itself? $\endgroup$
    – Rand al'Thor
    Commented Nov 17, 2014 at 21:46
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    $\begingroup$ Because I'm derp. Never spotted that horrid mistake since now.. Editing. Feeling so sorry.. :( $\endgroup$
    – Emi987
    Commented Nov 17, 2014 at 21:59
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    $\begingroup$ If I'd seen the question as it stands now, I would've got the answer (almost) immediately! $\endgroup$
    – Rand al'Thor
    Commented Nov 17, 2014 at 22:05
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    $\begingroup$ @randal'thor Congrats and thanks. Haha, I really did wrote the exact answer in my "answer". Seems like I just cannot get over with the prime^prime clue. Now it's clarified to be squared, I think I can rest in peace. $\endgroup$
    – user5508
    Commented Nov 18, 2014 at 2:35
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Here are my thoughts:

4 Characters, not 4 Numbers, so the password is:
22/7
It's the historical approximation of pi, and the date of the letter.
Bonus: the prime counting function is denoted as pi(x)

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  • $\begingroup$ +1 because you're so right! Can't believe I missed that. $\endgroup$
    – A E
    Commented Nov 13, 2014 at 19:43
  • $\begingroup$ You are on the right track my man! You are one step away from the solution! $\endgroup$
    – Emi987
    Commented Nov 13, 2014 at 20:22
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    $\begingroup$ @Emi987 I'm not following how this is related to n^n where n is prime... $\endgroup$
    – Michael
    Commented Nov 14, 2014 at 21:51
  • $\begingroup$ @Micheal There is one and only one. I'll give an hint soon! $\endgroup$
    – Emi987
    Commented Nov 14, 2014 at 22:01
  • $\begingroup$ @Michael: well that series is also called pi (no relation), but I'm not sure where that info takes us. $\endgroup$
    – A E
    Commented Nov 15, 2014 at 17:30
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Everything is related to $\pi$.

-22/7 is approximation of $\pi$ (tempted to think that it will be $\pi/14$, but $14$ is probably useless).
-24:00, which can also be written as $4\pi$ in radian (that's why you didnt write 00:00).
- 6-folded paper, which appears in $\zeta(2)=\pi^2/6$ (makes me wonder if paper's length is $\pi$).
- Primes are related to $\pi$ in many ways (not sure why powered to itself though).

That said, I still don't have a solution. Derp.

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  • $\begingroup$ How is 24:00 the same as 2pi?? $\endgroup$
    – Rand al'Thor
    Commented Nov 15, 2014 at 13:45
  • $\begingroup$ You indirectly wrote the solution, my friend. Don't be a derp, keep on going! Ps: 24.00 is just a stupind number not related to the solution btw :P $\endgroup$
    – Emi987
    Commented Nov 15, 2014 at 13:46
  • $\begingroup$ @randal'thor, my bad. Seems like 24 hours will require the hour hand to spin around the clock twice, which will be 4π. $\endgroup$
    – user5508
    Commented Nov 15, 2014 at 14:03
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    $\begingroup$ @Emi987. Am I right to say that "You can choose to stop here or keep on with all the prime numbers you can evaluate!" means that you are computing an infinite series that will approximate the answer, but computing up to the fifth term is good enough for the 4 characters solution? In that case, maybe we can compute pi^2/6 and get 1.64. But it still doesn't justify why prime^prime, hmmm... $\endgroup$
    – user5508
    Commented Nov 15, 2014 at 14:58
  • $\begingroup$ @nexolute You are absolute right, but you are keeping on going around the answer without spotting it. You got it, actually, but it's still not what I'm looking for! It's easier than the evaluation, trust me! I can tell you you spotted all the clues! $\endgroup$
    – Emi987
    Commented Nov 15, 2014 at 17:03
6
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My attempt is

3125

The reasoning:

Of the calculations listed, 5^5 is the only one that has an answer 4 digits long.

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It didn't matter what you type. The bomb won't go off anyway

24 Hour clocks run from 00:00.00 to 23:59.59 and then roll back over to 00:00.00. If the bomber has set the clock to detonate at 24:00 then the clock will never reach that number.

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  • $\begingroup$ The note doesn't say "I set the timer to 24:00", but "it will explode at 24:00", which is fine used in speech/writing where convenient. $\endgroup$
    – Set Big O
    Commented Nov 13, 2014 at 20:42
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did anyone notice the date of the message wrote at 22/7? and it will blow up at midnight, which will change the time to 23/7 00:00. since the topic is prime number, 23 and 7 are prime number too.

so i think that the answer is 23/7

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4
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I'm just going to try..

NONE, since anything raised to a number except 0 or 1 have multiples and cant be a prime number

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    $\begingroup$ I'm not sure I understand. $\endgroup$
    – A E
    Commented Nov 13, 2014 at 9:30
  • $\begingroup$ @AE I guees I'm wrong on understanding the question XD $\endgroup$
    – bluefire6
    Commented Nov 14, 2014 at 0:32
4
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Is the solution 1.006. Calculated as ζ(2), where we calculate for the next prime that is 13.

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The answer is 3125
5^5 is the only such equation with an answer of 4 digits.

4^4 = 256
5^5 = 3125
6^6 = 46656

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1
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Is it

zero, because that's what the reimann zeta function tends to, or zeros are kind of important in the study of it or something?

?

(no maths phd here, can you tell?)

or just

zeta

maybe?

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  • 1
    $\begingroup$ I guessed 'zeta' too, but that was wrong. The most famous open problem on the zeta function involves its zeros, but that seems a bit too far-fetched for this problem (and doesn't have anything to do with pi). $\endgroup$
    – Rand al'Thor
    Commented Nov 16, 2014 at 12:02
1
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One possibility is

0169

Since the paper was folded 6 times, that probably means that the password is

the square of the sixth prime

However, it took me around 20 seconds to get this answer, not 20 minutes, so perhaps this is not what you're looking for.

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  • $\begingroup$ Puzzle seems to say x^x, not x^2. $\endgroup$
    – A E
    Commented Nov 13, 2014 at 10:14

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