The Lazy Numbers - 24/03/2013

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The Lazy Numbers - 24/03/2013

Postby showmyiq » Sun Mar 24, 2013 10:05 am

The Lazy Numbers

The-Lazy-Numbers.gif
The Lazy Numbers
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Let’s define by lazy numbers, those numbers which are constructed by consecutive digits.
For example 45 is lazy number. 789 is lazy number. 456789 is lazy number too. 
Today I was analyzing the lazy numbers set, when I suddenly found such three-digit lazy number, which is exactly 2 less than a cube and 2 more than a square. By cube and square I define some random integer cube and square.
Can you found this number?
Can you prove it’s only one?
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Re: The Lazy Numbers - 24/03/2013

Postby DArk0n3 » Sun Mar 24, 2013 9:24 pm

I think the only solution is 123. All three-digit lazy numbers are of the kind abc, where a=x, b=x+1 and c=x+2
So we have 100x + 10(x+1) + x+2 = y^2 + 2 and 100x + 10(x+1) + x+2 = z^3 + 2. This means that 111x + 10 = y^2 and 111x + 14 = z^3, where x might take values from 0 to 9. After some trial and error (I tried finding a more clever way) I came to the conclusion that the only x that fulfills the the equalities is 1. So the three-digit number is 123.
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Re: The Lazy Numbers - 24/03/2013

Postby showmyiq » Sun Mar 24, 2013 9:59 pm

Absolutely correct answer!

Your name will be published as the one who cracked the puzzle. I hope you liked it!

Puzzle is closed!
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