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- An International Food Group consists of
twenty couples who meet four times a year for a meal. On each occasion, four
couples meet at each of five houses. The members of the group get along very
well together; nonetheless, there is always a bit of discontent during the year
when some couples meet more than once! Is it possible to plan four evenings
such that no two couples meet more than
once? There are many problems like this. They are called combinatorial
designs. Investigate others.
- What is the fewest number of colours needed to colour any map
if the rule is that no two countries with a common border can have the
same colour. Who discovered this? Why is the proof interesting? What
if Mars is also divided into areas so that these areas are owned by
different countries on earth. They too are coloured by the same rule
but the areas there must be coloured by the colour of the country they
belong to. How many colours are now needed? References:
[Hut], [Bal], [A&H].
- Discover all 17 ``different'' kinds of wallpaper. (Think about
how patterns on wallpaper repeat.) How is this related to the work of
Escher? Discover the history of this problem. References:
[Shep], [Cox], [C&C].
- Investigate self-avoiding random walks and where they naturally
occur. Reference: [Sla].
- Investigate the creation of secret codes (ciphers). Find out
where they are used (today!) and how they are used. Look at their
history. Build your own using prime numbers. References:
[F&K], [Bal].
- It is easy to cover a chessboard with dominoes so that no two
dominoes overlap and no square on the chessboard is uncovered. What
if with one square is removed from the chessboard? (impossible - why?)
What if two adjacent corners are removed? What if two opposite corners
are removed? (possible or impossible?) What if any two squares are
removed? What about using shapes other than dominoes (eg 3 squares joined together)? What about chessboards of different
dimensions? Reference: [Gol]. See the following
problem as well.
- Polyominoes are shapes made by connecting certain numbers of
equal-sized squares together. How many different ones can be made
from 2 squares? from 3, from 4, from 5? Investigate the shapes that
polynominoes can make. Play the ``choose-up'' Pentomino game.
References: [Gol], Volume 1 of [Gard3],
[Gard5].
- Find pictures which show that ; that ; and that . How many other ways can you find to prove these
identities? Is any one of them ``best''? References:
[S&R], or Proofs without words, regular feature of
Mathematics Magazine.
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