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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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Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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Sending a Parcel

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Catherine of Mount School, York and Joel of ACS Barker, Singapore realised that it is not necessary to calculate the lengths of the edges of the cuboid and they sent very similar solutions. This is Catherine's solution:

If the sides of the cuboid are x, y, and z and the areas of the rectangular faces are p, q and r then:
p = xy, q = yz and r = zx
It follows that pqr = (xy)(yz)(zx) = x 2 . y 2 . z 2 = (xyz) 2
So the volume = xyz = sqrt (pqr) = sqrt (3 x 12 x 25) = sqrt 900 = 30

Students from West Flegg Middle School, Norfolk and Madras College, St Andrew's and Russell Lower School, Ampthill also found the correct answer.