### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

### Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won \$1 200. What were the assets of the players at the beginning of the evening?

### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

# The Greedy Algorithm

##### Stage: 3 Challenge Level:

This problem follows on from Keep it Simple and Egyptian Fractions

So far you may have looked at how the Egyptians expressed fractions as the sum of different unit fractions. You may have started by considering fractions with small numerators, such as $\frac{2}{5}$, $\frac{3}{7}$, $\frac{4}{11}$, etc.
But how would the Egyptians have coped with fractions with large numerators such as $\frac{115}{137}$?

They might have written $\frac{115}{137} = \frac{1}{137} + \frac{1}{137} + \frac{1}{137}$....

and then used Jamie's method to make them all different, but this would have made an extremely lengthy calculation!

#### Fibonacci found an alternative strategy, called the Greedy
Algorithm:

At every stage, write down the largest possible unit fraction that is smaller than the fraction you're working on.

For example, let's start with $\frac{11}{12}$:
The largest possible unit fraction that is smaller than $\frac{11}{12}$ is $\frac{1}{2}$
$\frac{11}{12} - \frac{1}{2} = \frac{5}{12}$

So $\frac{11}{12} = \frac{1}{2} + \frac{5}{12}$

The largest possible unit fraction that is smaller than $\frac{5}{12}$ is $\frac{1}{3}$
$\frac{5}{12} - \frac{1}{3} = \frac{1}{12}$

So $\frac{11}{12} = \frac{1}{2} + \frac{1}{3} + \frac{1}{12}$

#### Choose a fraction of your own and apply the Greedy Algorithm to
see if you can finish up with a string of unit fractions.

Does the greedy algorithm always work?
Can all fractions be expressed as a sum of different unit fractions by applying the Greedy Algorithm?
Can you convince yourself of this?

Why do you think it is called the Greedy Algorithm? What do these words mean in a mathematical context?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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