thnik again!

Just wondering

- Sometimes I wonder.
- Me too!
- Just wonder. Are there intelligent lives on other planets?
- Are there intelligent lives on this planet?
- Why do we live?
- Why do people wonder why we live?
- Are you mocking me?
- I wonder what gave you that idea.
- Cut the crap!
- OK. Here is what I am really wondering about.
- Let’s hear it.
- Do squares with one million digits of which half are even exist?
- Now, that is interesting!
- Because it has a precise answer?
- Someone told me once that questions without answers are more frequent.
- I wonder what the ratio is.

Problems source: The Emissary Newsletter’s Puzzle section by Elwyn Berlekamp and Joe P Buhler.

This entry was posted on Wednesday, January 25th, 2012 at 12:04 am and is filed under Problem. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

6 Responses to “Just wondering”

Richard Sabey Says:
January 25th, 2012 at 7:33 am
Yes. Two examples coming up. Look at this sequence:

3² = 09
33² = 1089
333² = 110889
…
The 500,000th term is

3…3² = 1…108…89

with 500,000 even digits (a 0 and 499,999 8s) and 500,000 odd ones (499,999 1s and a 9).

Now this sequence:

6² = 36
66² = 4356
666² = 443556
…
The 500,000th term is

6…6² = 4…435…56

with 500,000 even digits (499,999 4s and a 6) and 500,000 odd ones (a 3 and 499,999 5s)
Richard Sabey Says:
January 25th, 2012 at 10:13 am
The smallest positive integer whose square has 1000000 digits is about 10^4999999.5, and the largest is 10^500000 – 1. So there are about (1- \frac{1}{\sqrt{10}})(10^{500000}) positive integers whose squares have 1000000 digits.

What’s the probability that, given that a number has 1000000 digits, half of them are even? Assuming (for simplicity) that each is randomly chosen, independently of the others, and the chance of it being even is 1/2, the chance is

$p = ({}_{1000000} C_{500000})/2^{1000000}$
$= \frac{1000000!}{(500000!^{2}2^{1000000}}$

Stirling’s approximation for factorials is

$n! \approx \sqrt{2N \pi}$

so, using N for 500000,

$p \approx \frac{(2N)!}{N!^2 2^{2N}}$
$\approx \frac{\sqrt{4N \pi}(2N)^{2N}e^{-2N}}{(\sqrt{2N \pi}N^{N}e^{-N})^{2}2^{2N}}$
$\approx \frac{\sqrt{4N \pi}2^{2N}e^{-2N}}{(\sqrt{2N \pi}e^{-N})^{2}2^{2N}}$ cancelling Ns
$\approx \frac{\sqrt{4N \pi}2^{2N}}{(\sqrt{2 N \pi})^{2}2^{2N}}$ cancelling e’s
$\approx \frac{\sqrt{N \pi}}{(\sqrt{N \pi})^{2}}$ cancelling 2′s
$\approx \frac{1}{\sqrt{N \pi}}$
so there are about
$\frac{(1-\frac{1}{\sqrt{10}})10^{500000}}{\sqrt{500000 \pi}}$
such integers.

Crossing my fingers that all that LaTeX works. (Jan, please could we have a preview of how our comment will look, as opposed to a preview of what’s in the type-in box?)
Jan Nordgreen Says:
January 25th, 2012 at 11:00 am
Richard, I will fix any latex that is not working, if I can.

I will try to find out if latex can be previewed.

Instead of previewing latex it would be nice to be able to edit your own comments. However, I guess that can’t be done easily.
Richard Sabey Says:
January 25th, 2012 at 7:05 pm
Not to worry, Jan. This morning I found http://www.codecogs.com/latex/eqneditor.php
which previews your LaTeX as you type it!
Jan Nordgreen Says:
January 25th, 2012 at 7:37 pm
Great! I will add a link to it in “How to use LaTeX in a comment.
David Brooks Says:
January 26th, 2012 at 1:46 am
9^2 = 81
99^2 = 9801
999^2 = 998001
9999^2 = 99980001
So the 500,000th term would have 499,999 “9″s.
Kewl!!!

Just wondering

6 Responses to “Just wondering”

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