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Door Security Protocol

Door Security Protocol:
Find the next number in the sequence
313, 331, 367, ...?
This is a test...

How long will it take you to work this out...

Quite a while is my guess.  And only a few will actually try to figure it out.  Most will give up, a few will search online, and a few will have already seen the sequence recognize it and either remember the answer (the next number in the sequence) or derive it with the sparse knowledge they remember about the sequence.

Spoiler Alert!  Let me give you the background knowledge.

You know what a Prime Number is - do you recognize that all the numbers in the list are prime?

Do you know what a Happy Number is?  Or it's inverse an Unhappy Number?  Well, this is a bit more obscure in the general knowledge.  So here is a spoiler...  all those numbers are happy!  They are Happy Primes!

A number is happy when the repeated practice of replacing the number by the sum of the squares of its digits eventually arrives at one.  Example:  13 >> (1^2) + (3^2) = 1 + 9 = 10;  10 >> (1^2) + (0^2) = 1!  Therefore, 13 is Happy. 

There are some obvious patterns within happy numbers e.g. both 13 and 31 are happy.

Happy Number derivation pathways

Unhappy Number derivation pathways

I just happen to love these happy/unhappy diagrams.

References:

How to find a happy number

Doctor Who clip from "42" (the title of the show) where this puzzle is seen for the space ship's door security protocol. (Spoiler Alert - the answer is in this clip.)


A list of Happy Primes

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