### Special Numbers

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?

### First Forward Into Logo 6: Variables and Procedures

Learn to write procedures and build them into Logo programs. Learn to use variables.

# Weekly Problem 35 - 2008

##### Stage: 3 Challenge Level:
$2$

Say a cone weighs $x$, a sphere weighs $y$ and a cube weighs $z$.

Then we have $$2x + y = z \quad \text{(1)}$$since two cones and a sphere together weigh the same as a cube.

Also $$y + z = 3 x \quad \text{(2)}$$since a sphere and a cube together weigh the as three cones.

Taking $y$ from each side of (1) (or taking a sphere from each side of the first scales) gives $$2x=z-y\quad \text{(3)}$$

Thento find out how much one cone weighs, we take equation (3) from equation (2) to give
\begin{eqnarray} 3x - 2x &=& (y+z) - (z-y) \\ x &=& y+z-z+y \\ &=& 2y \end{eqnarray}

So a cone weighs the same as two spheres.

This problem is taken from the UKMT Mathematical Challenges.

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