[QUIZ] Happy Numbers (#93)

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by Shane Emmons

Write a program that tells whether a given integer is happy. A happy number is
found using the following process: Take the sum of the squares of its digits,
and continue iterating this process until it yields 1, or produces an infinite
loop.

For example the number 7:

7^2 = 49
4^2 + 9^2 = 97
9^2 + 7^2 = 130
1^2 + 3^2 + 0^2 = 10
1^2 + 0^2 = 1

If a number is not happy than it is obviously unhappy. Now that you have this
program, what is the largest happy number you can find? What is the happiest
number between 1 and 1,000,000. I define the happiest number as the smallest
number that finds the most other happy numbers with it, i.e. 7 found four other
numbers (49, 97, 130, and 10) making it a rank 4 in happiness.

If you find all these examples trivial, write you program so that it will find
happy numbers in other bases such as base 2 or 16. From there you can extend the
program so that it finds happy bases (other than 2 and 4). A happy bases is a
base where all numbers are happy. Good luck.
 
W

William Crawford

James said:
Write a program that tells whether a given integer is happy. A happy
number is
found using the following process: Take the sum of the squares of its
digits,
and continue iterating this process until it yields 1, or produces an
infinite
loop.

For example the number 7:

7^2 = 49
4^2 + 9^2 = 97
9^2 + 7^2 = 130
1^2 + 3^2 + 0^2 = 10
1^2 + 0^2 = 1

If a number is not happy than it is obviously unhappy. Now that you have
this
program, what is the largest happy number you can find? What is the
happiest
number between 1 and 1,000,000. I define the happiest number as the
smallest
number that finds the most other happy numbers with it, i.e. 7 found
four other
numbers (49, 97, 130, and 10) making it a rank 4 in happiness.
Click to expand...

-Boggles-

Okay, I'm feeling stupid here. What MAKES that number be happy? The
nearest I can find is that if it's an infinite loop, it isn't happy. If
it's not a loop, and resolves to 1 eventually. (It has to resolve to 1,
or it's be a loop.) It's happy.

Is that right?
 
J

James Edward Gray II

Okay, I'm feeling stupid here. What MAKES that number be happy? The
nearest I can find is that if it's an infinite loop, it isn't
happy. If
it's not a loop, and resolves to 1 eventually. (It has to resolve
to 1,
or it's be a loop.) It's happy.

Is that right?
Click to expand...

Sounds right on to me.

James Edward Gray II
 
P

Peter Hickman

Problem is that from the quiz it states that you either get a 1 or an
infinite loop and that an unhappy number is "obvious". Which is a sign
that something has not been explained clearly. Given that the Wolfram
page was clearer than the quiz at to what constitutes happy don't be
surprised if some people go off down the wrong path.
 
K

knaveofdiamonds

One note of advice - the operator to use is actually ** not ^ as might
be expected:

irb> 7 ^ 2 # Gives 5
irb> 7 ** 49 # Gives 49

Cheers,
Roland
 
M

Matthew Moss

Now that you have this program, what is the largest happy number you can find?

(10 ** N) where N is anything. You name N, I'll say N+1.
 
M

Morton Goldberg

If a number is not happy than it is obviously unhappy. Now that you
have this
program, what is the largest happy number you can find?
Click to expand...

Clearly, there is no largest happy number. Any number of the form
10**n in base b is b-relative happy (with rank 1). I feel this makes
the stated question an ill-conditioned one.
What is the happiest number between 1 and 1,000,000. I define the
happiest number
as the smallest number that finds the most other happy numbers with
it, i.e. 7
found four other numbers (49, 97, 130, and 10) making it a rank 4
in happiness.
Click to expand...

This seems a better question to pursue than the first. Other
questions one might explore: how does rank vary as the numbers
increase? Does maximum rank grow as happy numbers get bigger? Is
there a pattern? Are there interesting statistics?

Regards, Morton
 
C

Chris Gehlker

Sorry, I should have clarified that a Happy number results in 1,
and an
Unhappy number results in an infinite loop.
Click to expand...

That's not quite what the cited article says. it says that happy
numbers result in an infinite loop on 1 and unhappy numbers result in
an infinite loop on the series: 4, 16, 37, 58, 89, 145, 42, 20,
4, ... and that there are no other possibilities.
 
M

Morton Goldberg

If a number is not happy than it is obviously unhappy. Now that you
have this
program, what is the largest happy number you can find?
Click to expand...

Clearly, there is no largest happy number. Any number of the form
10**n in base b is b-relative happy (with rank 1). I feel this makes
the stated question an ill-conditioned one.
What is the happiest number between 1 and 1,000,000. I define the
happiest number
as the smallest number that finds the most other happy numbers with
it, i.e. 7
found four other numbers (49, 97, 130, and 10) making it a rank 4
in happiness.
Click to expand...

This seems a better question to pursue than the first. Other
questions one might explore: how does rank vary as the numbers
increase? Does maximum rank grow as happy numbers get bigger? Is
there a pattern? Are there interesting statistics?

Regards, Morton
 
R

Roland Swingler

Sorry, should have been 7 ** 2 :)

Out of interest, any mathmaticians care to enlighten me as to whether
happy numbers are in any way "useful"?

Cheers,
Roland
 
J

James Edward Gray II

I think you meant 7 ** 2 = 49, whereas
7 ** 49 = 256923577521058878088611477224235621321607

But, since these are integers, we are better off computing n * n,
not n ** 2. There's really no point (in time efficiency) to explicitly
raising n to a power p unless n is not an integer or p is
relatively large.
Click to expand...

#!/usr/bin/env ruby -w

require "benchmark"

TESTS = 1_000_000
Benchmark.bmbm(10) do |results|
results.report("Exponent:") { TESTS.times { |n| n ** 2 } }
results.report("Multiply:") { TESTS.times { |n| n * n } }
end
# >> Rehearsal ---------------------------------------------
# >> Exponent: 1.370000 0.000000 1.370000 ( 1.372358)
# >> Multiply: 1.730000 0.010000 1.740000 ( 1.747647)
# >> ------------------------------------ total: 3.110000sec
# >>
# >> user system total real
# >> Exponent: 1.440000 0.000000 1.440000 ( 1.452441)
# >> Multiply: 1.760000 0.010000 1.770000 ( 1.775988)

James Edward Gray II
 
P

Phrogz

Interestingly, there doesn't seem to be a pattern as to the number of
happy numbers in a given range as the base changes:

(My results...might be wrong. :)
How many Happy numbers between 1..10000?
Base 2: 10000 happy numbers.
Base 3: 1988 happy numbers.
Base 4: 10000 happy numbers.
Base 5: 2571 happy numbers.
Base 6: 645 happy numbers.
Base 7: 162 happy numbers.
Base 8: 549 happy numbers.
Base 9: 627 happy numbers.
Base 10: 1442 happy numbers.
Base 11: 196 happy numbers.
Base 12: 24 happy numbers.
Base 13: 582 happy numbers.
Base 14: 93 happy numbers.
Base 15: 164 happy numbers.
Base 16: 2585 happy numbers.
Base 17: 253 happy numbers.
Base 18: 4154 happy numbers.
Base 19: 3647 happy numbers.
Base 20: 1616 happy numbers.
Base 21: 45 happy numbers.
Base 22: 17 happy numbers.
Base 23: 19 happy numbers.
Base 24: 9 happy numbers.
Base 25: 519 happy numbers.
Base 26: 377 happy numbers.
Base 27: 279 happy numbers.
Base 28: 6 happy numbers.
Base 29: 1730 happy numbers.
Base 30: 5266 happy numbers.
Base 31: 11 happy numbers.
Base 32: 11 happy numbers.
Base 33: 84 happy numbers.
Base 34: 192 happy numbers.
Base 35: 77 happy numbers.
Base 36: 50 happy numbers.
 
P

Phrogz

Although others are sure to have better results, I thought I'd post
mine (no spoilers). I did it a simple way first, and then another more
sophisticated way second. Nice speed improvement.

The below tests 1000 numbers for happiness in 33 different bases.

range = 1..1000
require 'Benchmark'
Benchmark.bm(20){ |r|
r.report( "Technique 1" ){
3.upto(36){ |base|
range.select{ |i|
# find out if 'i' is a happy number in the supplied base
}
}
}

r.report( "Technique 2" ){
3.upto(36){ |base|
range.select{ |i|
# find out if 'i' is a happy number in the supplied base
}
}
}
}

user system total real
Technique 1 19.282000 0.031000 19.313000 ( 19.359000)
Technique 2 2.140000 0.000000 2.140000 ( 2.156000)
 
M

Matthew Moss

Out of interest, any mathmaticians care to enlighten me as to whether
happy numbers are in any way "useful"?
Click to expand...

As has been said, "real mathematicians" shudder that their work might be useful.

In actuality, sometimes things like this aren't, and sometimes they
are years later. Prime numbers never had much use until recent years,
where they have become very important for encryption.

Maybe happy numbers have a use... just no one knows it yet!
 

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