```On 3/5/07, M. Edward (Ed) Borasky <znmeb / cesmail.net> wrote:
> Yannick Grams wrote:
> > Hello to everyone!
> >
> > I have a mathematic problem I need to solve, and it involves finding
> > the values of three consecutive numbers, x, y and z. I need to be able
> > to find every instance of these from negative one million (-1,000,000)
> > up to one million (1,000,000). They must follow the following equations:
> >
> > x + y + z = 5
> > x + y - z = 7
> > (x - y)(x - y)(x - y)  + (y - z)(y - z)(y - z) = (x - z)(x - z)(x - z)
> >
> > I'm completely and utterly stumped.

> Well, one of the reasons you might be stumped is that the problem as
> posed does not appear to have a solution. I punched it in to Maxima:
>
> (%o1) z+y+x=5
> (%o2) -z+y+x=7
> (%o4) (y-z)^3+(x-y)^3=(x-z)^3
> (%i5) solve([%o1,%o2,%o4], [x,y,z]);
> (%o5) [[x=-1,y=7,z=-1],[x=3,y=3,z=-1],[x=7,y=-1,z=-1]]
>
> Since one of the equations is a cubic, there are three solutions, and
> they are all smallish integers. But they are *not* consecutive!
>
> Are you sure about the problem statement?

You can tell there's no solution for consecutive X, Y, Z just from
looking at that last equation, each side boils down to a different
constant:

Since the numbers are consecutive
y = x + 1
z = y + 1

Then:
x - y = x - (x+1) = -1
y - z = y - (y+1) = -1
x - z = x - (y+1) = x - (x+1+2) = -2

so:

(x-y)(x-y)(x-y) + (y-z)(y-z)(y-z) =
(-1)(-1)(-1) + (-1)(-1)(-1) =
-2

but:

(x - z)(x - z)(x - z) =
(-2)(-2)(-2) = -8

--
Rick DeNatale

My blog on Ruby