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Shiny Numbers (Posted on 2013-06-20) Difficulty: 3 of 5
A positive integer is called shiny if it can be written as the sum of two not necessarily distinct integers a and b which have the same sum of their digits. For instance, 2012 is shiny, because 2012 = 2005 + 7, and both 2005 and 7 have the same sum of their digits. Find all positive integers which are not shiny (the dark integers).

No Solution Yet Submitted by Danish Ahmed Khan    
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First steps to proof | Comment 5 of 6 |
Rather than prove which numbers are dark, it may be easier to prove which numbers are shiny:

First all even numbers are shiny: 2n = n+n which is specifically allowed. 

It shouldn't be too hard to prove every number ending in 1 is shiny, (but I'm having a bit of trouble if the number ends in 91 or 01.) 

Same with ending in 3,5,7.

It may be helpful to note: If x is shiny and y is an n-digit shiny number then x*10^n + y is shiny.  This may help to explain the form of the dark numbers.

  Posted by Jer on 2013-06-21 10:05:40
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