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 $1 200. What were the assets of the players at the beginning of the evening? Plutarch's Boxes According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes? Diminishing Returns Stage: 3 Challenge Level: We received several good answers to this problem. Daisy from Ricards Lodge School For the first part I got$\frac{21}{64}$I followed a very basic method which involved drawing a 4 by 4 grid on the image. When I did that I realised that the blue outer triangles (with the grid lines now drawn) was 1/4 of the image's area. I then worked my way inwards and then realised that the next set of blue triangles formed a square that fitted in one of my grid squares ($\frac{1}{16}$th) of the image. The next and final blue triangles formed a square which was very small. To work out how big it was I had to draw a final grid in a 4 by 4 square. I cut it into quarters and realised that was the size of my final blue square which menas that it was$\frac{1}{64}$th of the image. Finally I added up my fractions:$\frac{1}{4}$+$\frac{1}{16}$+$\frac{1}{64}$=$\frac{21}{64}$Charlotte, Kathryn and Tess from Christ the King Catholic High School added up the areas occupied by different colours: We decided to draw out the square with the blue triangles in it. Then we found out what fraction the different sized triangles came to. The 4 larger blue triangles came to$\frac{1}{4}$. The 4 smaller blue triangles came to$\frac{1}{16}$. The 4 smallest blue triangles came to$\frac{1}{64}$. We added up all the fractions which came to$\frac{21}{64}$. We changed this to a percentage which gave us 32.8125%. Matthew from The King's School in Grantham went about it differently: If the first pink square (in the middle) has an area of 1, the first blue square will have an area of 2, with half of the blue square coloured blue. The second pink square will have an area of 4, so the second blue square will have an area of 8, again with half of the blue square coloured blue. The third pink square will have an area of 16 and the third blue square will have an area of 32 - with half of the square coloured blue.  AREA OF BLUE SQUARES FRACTION COLOURED BLUE TOTAL AREA 2$\frac{1}{2}$1 8$\frac{1}{2}$4 32$\frac{1}{2}$16 21 The largest pink square will have an area of 64, so the area shaded blue takes up$\frac{21}{64}$of the total area. Natasha from Ricards Lodge School realised that the blue areas added up to about$\frac{1}{3}$of the total area: Assume central pink square = area x The central pink square has an area$\frac{1}{2}$the size of the next largest set of pink shaded triangles (2x). This set has an area$\frac{1}{4}$the size of the next largest set (8x), which in turn has an area$\frac{1}{4}$the size of the largest set (32x). In total this comes to 43x. The smallest set of 4 blue shaded triangles also has an area x. The smalllest set has an area$\frac{1}{4}$the size of the next largest set (4x),which in turn has an area$\frac{1}{4}$the size of the largest set (16x). In total this comes to 21x. Roughly twice as much area is pink as is blue, therefore approximately$\frac{1}{3}$of the diagram is blue. John, also from King's, noticed that: By looking at the different layers you can see that the area of one set of 4 triangles is equal to the total area of every layer within: Layer 1$\frac{1}{2}$Pink Layer 2$\frac{1}{4}$Blue Layer 3$\frac{1}{8}$Pink Layer 4$\frac{1}{16}$Blue Layer 5$\frac{1}{32}$Pink Layer 6$\frac{1}{64}$Blue Layer 7$\frac{1}{64}$Pink Then we added the area of blue together:$\frac{1}{4}$+$\frac{1}{16}$+$\frac{1}{64}$=$\frac{21}{64}$Chan also added up the areas. Harry from West Buckland explained it like this: In the first image,$\frac{21}{64}$or 32.8125% is covered by blue. This is because the largest pink triangles take up 50% of the image, leaving 50% for the remaining image. 50% of this remaining area is taken up by the biggest blue triangles, leaving 25% for the rest of the image. This continues, leaving$\frac{1}{4}$of the image covered by the largest blue triangles,$\frac{1}{16}$by the second largest and$\frac{1}{64}$by the smallest - each takes$\frac{1}{4}$of the area the previous triangle has taken. When$\frac{1}{4}$,$\frac{1}{16}$and$\frac{1}{64}$are added, the total is$\frac{21}{64}$(32.8125%). Max from St Martin's School observed that each square is half the size of the previous one. This is how Ella from Ricards Lodge School explained it: The percentage of the square that is shaded blue is 32.8125%. This is because the outsides of the square (which is pink) can fold inwards to cover the inside colours, which means that the outside pink triangles are equal to 50%, so the inside is 50% of the shape as well. This then means that the biggest blue triangles can fold inwards to cover the inner square of that, which would mean those triangles equal 25% of the overall shape. You then continue to imagine folding each set of triangles inwards, getting 12.5% for the next pink triangles and 6.25% for the next blue triangles and 3.125% for the next pink triangles and 1.5625% for the last blue triangles and also 1.5625% for the final pink square. But if the square was to keep going on with the triangles continuing in the same pattern the answer would head towards 1 third blue. This is because if you continued halving and halving you would have 2 thirds pink. Yoram expanded on this, saying that The blue part is one third. Because the largest four blue triangles make two of the biggest pink triangles. And it is the same all again with the smaller triangles. And so, pink is twice as much as blue. James from Wilson's School correctly deduced that this meant that the blue made up one third of the total area. You can work this out by simply trying to disect the pattern and then making a pile of pink and blue triangles. If you do this, you will find there is a larger proportion of pink than blue. The ratio of pink to blue is 2:1. This can be written as... the pink fraction of triangles =$\frac{2}{3}$and the blue fraction of triangles =$\frac{1}{3}$. Ben from St Martin's School wrote out a table showing the fractions of the whole area that each new set of squares represented: In order to solve the problem, you must first work out that the blue triangles are half of the pink triangles surrounding them. I have devised a table of how the fractions work as the squares get smaller: Pink Fraction:$\frac{1}{2}$,$\frac{1}{8}$,$\frac{1}{32}$,$\frac{1}{128}$... Blue Fraction:$\frac{1}{4}$,$\frac{1}{16}$,$\frac{1}{64}$,$\frac{1}{256}$... Ben explained that if you take an equal number of terms from each of these sequences, blue always makes up$\frac{1}{3}\$ of the total.

Do you agree that this trend will continue?

Can you think of any other ways to do this problem?

Simone from Harris Academy in Merton suggested what happens in a couple of other images:

For the one that is numbered 1 and the one numbered 9, the solution is that for every triangle that is blue, there is a purple area that is half as big. The ratio is 2:1

Niharika from Leicester High School for Girls produced this clear algebraic analysis of the situation. Well done to you all.