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

ganesh
Moderator
Registered: 2005-06-28
Posts: 13,227

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.

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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: 13,227

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


Character is who you are when no one is looking.

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