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# Maths Filler

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Age 11 to 14

Challenge Level

A series of vessels of uniform cross section look like letters when viewed face-on. They are 1cm thick, and the corners of the vessels have either whole or half cm values for their coordinates.

Water is poured slowly into each vessel through **one** of the holes at the top at a rate of 1 cm$^3$ per minute.

Seven of these vessels take the same time to fill up. Which are they?

Which one takes the longest to fill?

Which would fill up first?

The height of the pool of water in one of the vessels is measured and a chart of the height against time is plotted, as follows:

Which vessel does this chart correspond to?

Can you explain what each part of the chart corresponds to?

Can you make charts of height against time for some of the other letters?

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?

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal spoons. Each day a spoonful was used to perfume the bath of a beautiful princess. For how many days did the whole jar last? The genie's master replied: Five hundred and ninety five days. What three numbers do the genie's words granid, ozvik and vaswik stand for?