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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.

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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?

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Take Ten

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3 cube?

Changing Areas, Changing Volumes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2
This problem follows on from Changing Areas, Changing Perimeters.
 
 
Here are the dimensions of nine cuboids. You can download a set of cards here.

 
 
1 by 2 by 28
cuboid  
 
 
4 by 4 by 4
cube
 
 
2 by 4 by 7
cuboid  
 
 
1 by 2 by 26
cuboid  
 
 
2 by 4 by 6
cuboid  
 
 
4 by 5 by 6
cuboid  
 
 
4 by 5 by 7
cuboid  
 
 
1 by 2 by 24
cuboid  
 
 
1 by 4 by 14
cuboid  
 

 
The challenge is to arrange them in a 3 by 3 grid like the one below:
 area and volume grid
 
 
As you go from left to right, the surface area of the shapes must increase.
As you go from top to bottom, the volume of the shapes must increase.
All the cuboids in the middle column must have the same surface area.
All the cuboids on the middle row must have the same volume.

What reasoning can you use to help you to decide where each cuboid must go?
 

Once you've placed the nine cards, take a look at the extended grid below:
 extended grid
The ticks represent the nine cards you've already placed. Can you create cards with dimensions for cuboids that could go in the four blank spaces that satisfy the same criteria? 
 
 
Can you design a set of cards of your own with a different cuboid in the centre?