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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?
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?
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.
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.
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.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.