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#1 2005-08-31 16:11:07
- ganesh
- Moderator
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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.
#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
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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.