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

ganesh
Moderator
Registered: 2005-06-28
Posts: 21,104

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: 21,104

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 smile


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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