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Go for short! (Posted on 2005-04-19) Difficulty: 2 of 5
Looking at the "Square of an Odd" puzzle that asks to prove that the square of an odd number is always 1 more than a multiple of 8, a professor gave this four parts proof: "All odd numbers are of the form 8K+1, 8K+3, 8K+5 or 8K+7. Squaring these numbers produces 8M+1, 8M+9, 8M+25 or 8M+49, which are all of the form 8N+1. QED"

Another professor came by, and gave a single line proof. Can you manage it?

Note: no one who answered the original problem produced either the four parts solution, or the single line one.

See The Solution Submitted by e.g.    
Rating: 2.0000 (2 votes)

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"computer" solution | Comment 4 of 7 |

Any odd number, when written in binary, is ...fedcb1, where b, c, d, e, and f are the multiples of the powers of 2, 4, 8, 16, and 32, respectively.  (The number is equal to 1 + 2b + 4c + 8d + 16e + ...)

Squaring this number will yield 1 + 4b + 8c + 4bb + 16d + 8bc + 32e + 16bd + 16cc + ... , which equals 1 + 8k for all {0,1} values of b,c,d,e,f,g,...


  Posted by Duncan Garp on 2005-06-04 05:34:12
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