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## #1 2005-08-30 18:11:07

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,730

### 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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #2 2005-08-31 19:35:46

wcy
Member
Registered: 2005-08-04
Posts: 117

### Re: 2n ones and n twos

wow this is amazing

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## #3 2005-09-01 06:54:18

Roraborealis
Member
Registered: 2005-03-17
Posts: 1,594

### 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?

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## #4 2005-09-01 20:05:43

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,730

### 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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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