## Monday, July 6, 2015

### Fill up grids such that $\dfrac{\square}{\square \square}+\dfrac{\square}{\square \square} +\dfrac{\square}{\square\square}=1$

In the following grid, fill up the numbers from 1 to 9 (without repetition) such that their sum is $1$.

$\dfrac{\square}{\square \square}+\dfrac{\square}{\square \square} +\dfrac{\square}{\square\square}=1$

Solution provided by Mark, another contributor of this blog:

I would begin with:

[MATH]\frac{9}{12}[/MATH]

since this is the largest addend possible. This leaves [MATH]\frac{1}{4}[/MATH].

Observing that:

[MATH]\frac{1}{4}=\frac{17}{68}=\frac{2\cdot5+7}{68}= \frac{5}{34}+\frac{7}{68}[/MATH]

We now have:

[MATH]\frac{9}{12}+\frac{5}{34}+\frac{7}{68}=1[/MATH]