Problem 45250. Be happy

Created by Asif Newaz
Appears in Number theory

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check whether the given number is happy in b-base.

A happy number can be defined as a number which will yield 1 when it is replaced by the sum of the square of its digits repeatedly. If this process results in an endless cycle of numbers containing 4, then the number is called an unhappy number.

This is the case for base-10. For other bases, different scenerios would occur.

https://en.wikipedia.org/wiki/Happy_number

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Last Solution submitted on Feb 23, 2025

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Ned Gulley on 3 Jan 2020

Hi Asif: Can you give a little more detail to your solvers about what "happy" and "b-base' mean? Maybe link to Wikipedia reference pages?

Asif Newaz on 3 Jan 2020

https://en.wikipedia.org/wiki/Happy_number

William on 5 Jan 2020

Asif: If I am reading that reference correctly, then test suite problems 4 and 7 are incorrect. In fact, problem 4 has base b=4, and according to the reference b=4 is a "happy base" in which *all* numbers are happy.

Asif Newaz on 5 Jan 2020

thank u william for pointing that out.
i've modified the test suites.
kindly inform if u find any more discrepancies.

William on 6 Jan 2020

Asif: Yes, I am now seeing a discrepancy with problem 6, but all the others agree with my code.

Asif Newaz on 6 Jan 2020

william, so far i've understood the concept, test-6 is correct.
https://en.wikipedia.org/wiki/Perfect_digital_invariant
have a look..I think it'll make things clear
but still if i'm wrong,do inform

William on 6 Jan 2020

Well, it appears to me that the number 742356 is a pre-periodic point of the perfect digital invariant n=8 in base 3. However, since the perfect digital invariant is n=8 rather than n=1, this is not a happy number. At least, that is how I read it!

Asif Newaz on 8 Jan 2020

william
did u look into the nontrivial PDI portion?
i might have been wrong-i 'll look into it when i get some time. i'm a bit busy now with my new job

William on 10 Jan 2020

Asif, Yes, I looked at the information on non-trivial perfect digital invariants. Near the end of that reference, there is a short section entitled "Relation to happy numbers" that indicates that in order for a number to be happy, it needs to be a perfect digital invariant with the value of 1. In this case, the invariant is 8 (or 22 in base-3), so I interprete that as saying that it is a non-trivial PDI, but not happy.

Asif Newaz on 11 Jan 2020

william,
thanks man.. sorry i didn't go through all that info.
i've updated the problem.it should be okay now

Jan Olsen on 19 Aug 2020

Is test 10 wrong? n=3148 in base 13 cylces 79-A0-79-A0, not a happy number.

Are Mjaavatten on 26 Apr 2021

Jan Olsen is right and test 10 is definitely wrong. In iteration 5 most solvers (including Asif, it seems) translate dec2base(10,13) = ‘A’ to digital by ‘A’-‘0’ = 17. The correct is of course 10, leading to the cycle noted by Jan. The test suite should be corrected.

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Problem 45250. Be happy

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