## The Lazy Numbers - 24/03/2013

### The Lazy Numbers - 24/03/2013

The Lazy Numbers

The Lazy Numbers
The-Lazy-Numbers.gif (49.47 KiB) Viewed 19965 times

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?

showmyiq

Posts: 390
Joined: Sat Sep 15, 2012 9:45 pm

### Re: The Lazy Numbers - 24/03/2013

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.
DArk0n3

Posts: 76
Joined: Sat Oct 06, 2012 4:20 pm

### Re: The Lazy Numbers - 24/03/2013

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

Puzzle is closed!

showmyiq