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 Cantor Set

Stage: 3 Challenge Level:

Liam from Wilbarston School sent us this well-explained solution:

Only the middle thirds of line segments are removed, so endpoints of line segments will not be removed.

Therefore, all I need to do is determine which are endpoints and which are not.

$\frac{1}{3}$ is an endpoint of C2, so it is in the Cantor set.

$\frac{3}{81}$ = $\frac{1}{27}$

$\frac{1}{27}$ is an endpoint of C4, and therefore $\frac{3}{81}$ is in the Cantor set.

$\frac{4}{9}$ is between $\frac{1}{3}$ and $\frac{2}{3}$, thus it is in the middle third of C1 and is removed from C2 and as such isn't in the Cantor set.

$\frac{4}{81}$ is between $\frac{1}{27}$ and $\frac{2}{27}$ and therefore is in the middle third of C3.

It is removed from C4 and therefore does not belong.

Famous mathematicians. Fractal. Interactivities. Visualising. Fractions. Generalising. Working systematically. Infinity. Cubes. Mathematical reasoning & proof.

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F'arc'tion

Do Unto Caesar

Plutarch's Boxes

The Cantor Set

Stage: 3 Challenge Level: