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## #1 2005-08-31 16:11:07

ganesh
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### 2n ones and n twos

I noticed this when I was browsing the net for interesting Mathematics.
I liked this proof, maybe you like it too!

Write, side by side, the numeral 1 an even number of times. Subtract from the number thus formed the number obtained by writing, side by side, a series of 2s half the length of the first number. You will always get a perfect square. For instance,
1111 - 22 = 1089 = 33²
Can you say why this is?

11...1  -  22...2 =  11...1 11...1  - 2(11...1)
------     ------    ------ ------      ------
2n times   n times   n times n times    n times

=  11...1 00...0  -   11...1
------ ------      ------
n times n times    n times

=  11...1 x (100...0 - 1)
------     ------
n times    n times

=  11...1 x 99...9
------   ------
n times   n times

=  11...1 x 9 x 11...1
------       ------
n times       n times

=  3²  x 11...1²
------
n times

=         33...3²
------
n times

Character is who you are when no one is looking.

## #2 2005-09-01 17:35:46

wcy
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### Re: 2n ones and n twos

wow this is amazing

## #3 2005-09-02 04:54:18

Roraborealis
Super Member

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### Re: 2n ones and n twos

Why does that work?

School is practice for the future. Practice makes perfect. But - nobody's perfect, so why practice?

## #4 2005-09-02 18:05:43

ganesh
Moderator

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### Re: 2n ones and n twos

Because, the resultant is always 3² or 33² or 333² or 3333² etc.
Follow every step of the proof carefully, you can understand the reasoning

Character is who you are when no one is looking.