# Happy Numbers (#93)

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by Shane E.

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.

James G. wrote:

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.

-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?

On Sep 1, 2006, at 8:12 AM, William C. wrote:

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?

Sounds right on to me.

James Edward G. II

Peter H. wrote:

Doesn’t this mean that all numbers are happy?

Do it by hand for 3…

Doesn’t this mean that all numbers are happy?

3^2=9
9^2=81
8^2+1^2=65
6^2+5^2=61
6^2+1^2=37
3^2+7^2=58
5^2+8^2=89
8^2+9^2=145
1^2+4^2+5^2=42
4^2+2^2=20
2^2+0^2=4
4^2=16
1^2+6^2=37
We just started a loop here since 37 has already been found.

James G. wrote:

On Sep 1, 2006, at 8:41 AM, Peter H. wrote:

Doesn’t this mean that all numbers are happy?

Here’s some further reading for those still confused:

http://mathworld.wolfram.com/HappyNumber.html

James Edward G. II

Wow. Lots of strategy on that page. Almost like cheating

On a side note… Did anyone NOT calculate 3 by hand? Heh.

On Sep 1, 2006, at 8:41 AM, Peter H. wrote:

Doesn’t this mean that all numbers are happy?

Here’s some further reading for those still confused:

James Edward G. II

On 9/1/06, William C. [email protected] wrote:

On a side note… Did anyone NOT calculate 3 by hand? Heh.

/me sheepishly raises his hand

I didn’t use my hands, just my head.

Jacob F.

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

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

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.

knaveofdiamonds wrote:

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

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.

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.

On Sep 1, 2006, at 7:58 AM, Shane E. wrote:

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

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.

Peter H. wrote:

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.

As I understand the Wolfram article, an unhappy number never hits 1 as a
result, whereas a happy number eventually hits 1:

“Unhappy numbers have eventually periodic sequences of s(sub)i which
never
reach 1.”

The quiz item emphasizes a criterion that is only mentioned in passing
on
the Wolfram page, that is, there are degrees of happiness, and a number
that has many happy predecessors is, umm, more happy. So a relatively
large
number that eventually results in 1, but with a lot of steps along the
way
(therefore more happy ancestors), is happier.

On Sep 1, 2006, at 9:01 AM, Ruby Q. wrote:

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?

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.

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

On Sep 1, 2006, at 9:01 AM, Ruby Q. wrote:

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?

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.

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

On 1-Sep-06, at 11:00 AM, knaveofdiamonds wrote:

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

I hope that’s not cut & pasted! 7 ** 49 is a little bigger:

ratdog:~ mike\$ irb --prompt simple

7 ^ 2
=> 5
7 ** 2
=> 49
7 ** 49
=> 256923577521058878088611477224235621321607

Mike

The “`Stok’ disclaimers” apply.

On Sep 1, 2006, at 10:40 AM, Paul L. wrote:

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.

#!/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

# >> Multiply: 1.760000 0.010000 1.770000 ( 1.775988)

James Edward G. II