Activities
You have 12 cubes, each with sides 1 cm long. How many different cuboids can youmake using all the cubes for each one?
Two are shown here, but these are essentially the same, and could be described as
3 × 2 × 2 cuboids. There are 3 other different cuboids that can be made from 12
How many different cuboids can you make using:
Without drawing them or using cubes, decide how many different cuboids you couldmake using the following numbers of cubes:
Can you determine a general result which gives you the number of different cuboids it ispossible to make possible using any number of cubes?
(Hint: write each number as a product of its prime factors and look at the sum of the
Collect data from as many cereal boxes as possible, when you next visit a supermarket.
Determine the volume that each one contains by measuring the box, and the mass of itscontents (printed on box). Calculate the density of the contents of each box, and completea table like the one below:
Comment on your results; for example, decide which is the most dense cereal, and whichthe least dense.
You might like to consider the following questions:
Do all brands of the same type of cereal have the same density?
Do large packets of cereal have the same density as smaller packets of the samecereal?
In each case, determine the cost per gm of the cereal. Are the larger boxes better valuethan the smaller ones?
If you are given 27 cubes, each with sides 1 cm long,
describe the 3 cuboids you can make, using all 27 of the cubes,
which of these cuboids has the smallest surface area?
For example, this cuboid is made from 27 cubes:
Describe cuboids that have the smallest surface area that can be made from:
40 cubes, each with sides 1 cm long.
Describe how to calculate the smallest surface area for a cuboid made from acertain number of cubes.
The Danish puzzle expert Piet Hein invented the activity below, consisting of 7 pieces –one piece contains 3 unit cubes, whilst all the others include 4 unit cubes. The method ofconstruction is shown below:
These 7 pieces combine together (in 240 different ways!) to form a 3 × 3 × 3 cube . Before attempting the cube construction, try the two problems below. First, though,make all the 7 pieces, using 27 cubes.
Combine any two pieces to form this structure:
Use all seven pieces to make the two shapes below:
Make a 3 × 3 × 3 cube, using all 7 pieces. Notes and solutions are given only where appropriate.
General results can be deduced from the factorisation into prime numbers,,
Note that this does not always give the exact answer, but it does give a lower bound. In some cases, the lower bound is the exact answer, but note, for example, that:
⇒ 2 + 2 = 4 cuboids, but there are 6 possible cuboids,
⇒ 7 cuboids, but there are, in fact, 8 possible cuboids.
Minimum for shape that is a cube or closest to a cube.
Note that 8 unit cubes are needed for this shape, so that the No. 1 piece (whichhas 3 unit cubes) is not suitable.
The shape on the left uses pieces 2, 4 and 5. Note that it is possible, using otherpieces, to form the shape on the left, and then be unable to make the shape onthe right.
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