Can you explain the strategy for winning this game with any target?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
Can you explain how this card trick works?
This problem demonstrates that multiple,
different methods can be used to approach a particular question. In
addition, you are encouraged to devise a general formula for the
number of chocolates in any hexagonal box. In doing this, it then
becomes very straightforward to work out the number of chocolates
there are, however large the box is.
Working out general formulae is not only fun
and satisfying, but is also very useful; it provides a quick way to
extend the original set of data, without having to work out every
consecutive point. Scientists, Mathematicians, Engineers,
Accountants...they all find general formulae very helpful, and make
use of them regularly.
In the Christmas Chocolates problem, the
formula can be extended to any size of box, because the differences
are uniformly spaced. However, sometimes we do not have "perfect"
information. For example, an experiment may yield slightly
different results each time (this is why we do repeat experiments),
but we cannot do an infinite number of trials. Thus we settle on an
appropriate number, and then use the data collected to see if there
is a pattern. From this, perhaps with more experiments, we may be
able to devise a general formula. This can then be used to make
predictions, and then experiments can be used to test these. The
formula may then be refined, and then new predictions made, and so
on. Remember though that the formula may not be true for all
conditions, unlike the chocolate problem. For example, the formula
may only apply within a given temperature range. Nevertheless,
having a formula is very useful; it not only makes calculations
more efficient, but can also yield great insight into the
properties of the problem.
Lots of great solutions were submitted,
so thank you to everyone for this! Many students noticed the shapes
that were made once Penny, Tom, and Matthew had eaten some
chocolates. They then realised that several of these shapes could
fit into the whole hexagonal box. This is useful as it provides a
convenient way of grouping the chocolates together, and therefore
quickens calculations. The patterns can then be used to extend to
other box sizes, as several students did.
Courtney, from Penrhos College,
submitted this solution. It is
lovely: clear, and illustrated. Lauren, from Tudhoe Grange, also
submitted a good solution. This solution is
colourful, and demonstrates the different shapes created by Penny,
Tom, and Matthew very clearly. Adam, from Wilson's School
Luke, from Wilmslow High School, used
the alternative method suggested by Adam. In
his solution, he added the rows of the
hexagon up to find out the total number of chocolates in the
Monty, from The Perse School, and Claudia
& Martha from Impington Village College noticed that Penny and
Tom's methods were extremely similar; Tom ate twice as much as
Penny, so the only difference between the caluclation is a factor
Philip, from Wilson's School extended the
methods used by Penny, Tom, and Matthew to work out the number of
chocolates in a box with ten along each edge:
Kevin, Leif, Oliver, Nisha, Olivia, Thomas,
and Henry from Highgate School explained the latter point made by
Hannah and Phil (about sharing the chocolates equally). They used
Matthew's method as an example:
They then applied the same principle to work
out the size of chocolate box the boys will be able to share out
equally (i.e. between two people):
In fact, as Philip from Wilson's School
What's the moral of this, then? Well, if
you want to share a box of chocolates equally between two or three
people, it should not be a hexagonal
box like the one in the question. You could, of course, not share
it equally... and have more for yourself... Even better, give more
to somebody else!
Thank you to everyone who submitted
solutions to this tasty chocolatey problem. If you enjoyed this
(and we know that you did!), have a go at Summing Squares and Picture Story. These
problems are nice extensions of this chocolate problem. If you
would like to try a similar, related problem for more practice,
try Picturing Triangle Numbers, Mystic Rose,