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A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

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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 Genie in the Jar

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?

Maths Filler

Age 11 to 14
Challenge Level
Well done to Anurag and Christina for their solutions to this problem:

The seven letters that take the same time to fill up are: I,L,O,E,M,T and H, all with a volume $14$cm$^3$ and thus taking $14$ minutes to fill.

The letter S takes the longest to fill up ($14.5$cm$^3$, $14.5$ minutes to fill).
The letter V fills up first ($13$cm$^3$, $13$ minutes to fill).
The letter A will take $13.5$ minutes to fill.

The graph corresponds to the letter M. Some points in the graph are over measured, for instance, the points between 4 and 5 cm.

Letter M

1. 0 - 3 minutes - filling one 'leg' of M, with rate of height increase constant due to constant width

2. 3 - 7 minutes - further water will run over in to the central dip of the M, and then once this is filled into the opposite leg. These have a combined volume of $4$cm$^3$ and so take 4 minutes to fill

3. 7 - 11 minutes - water fills top rectangular section with constant rate of height increase

4. 11 - 14 minutes - filling up top two trapezoidal sections of M

5. 14 - 16 minutes - letter completely full - no further height gain

Section 4 in the period 11-14 minutes should actually be represented by a curved line on the chart as the width of the section being filled is changing with height, and therefore so will the rate of height increase.