224 Unit 5 Advanced problem solving
putting oranges in the ‘dimples’ in the layer
below until no more layers can be made.
This is shown below left for a box of
16 (4 × 4) oranges on the bottom layer.
Clearly a 2 × 2 box would contain 5 oranges
(4 on the bottom and 1 above). How many
oranges would a 5 × 5 box contain? Can
you generalise for any square box?
What would happen with a rectangular
box? Start with a box containing 4 × 5
oranges on the bottom layer. Can you
develop rules which would allow you to
calculate the number of oranges stacked
in any rectangular box?
3 Milly is running a game at her school fête
to raise money for the school. Her idea is
to get people to throw two dice. The players
pay $1 per game and they win $2 if the two
numbers they throw differ by more than 2.
If 200 people play the game, how much
money will she expect to raise?
She is worried that people may be able
to calculate the odds for this game easily,
and that this may discourage them from
playing. What alternative criteria could
she consider for a win? What about, for
example, the product of the numbers on
the two dice or the two values written as
a two-digit number (e.g. 2 and 5 become
25)? In each case you think of, work out
the criterion for a win to ensure that she
makes a similar profit to that calculated
above. Look at these and any other
possibilities you may think of, calculating
the odds of winning for different rules.
You could also play the game as a class
activity and see whether the experimental
odds match the calculated value.Answers and comments are on page 330.1 Coins in most of the world’s currencies are
based on a decimal system, the individual
coins (below $1) being, for example, 1¢,
2¢, 5¢, 10¢, 20¢ and 50¢ (some may
also include a 25¢ coin). Consider a single
transaction to buy one item.
a Starting from a purchase worth 1¢, up
to what amount can such a transaction
be carried out using only one or two
coins? This could involve the purchaser
paying the exact amount with two coins,
or the purchaser offering one coin and
receiving one as change (for example,
an item costing 3¢ can be purchased
by offering a 5¢ coin and receiving a 2¢
coin in change).
b Can you develop an alternative coin
system which uses relatively few
coins but can make a big range of
values using only one or two coins?
For example, consider a coin system
starting with 1¢, 3¢, 5¢ and so on.
This investigation is potentially open-
ended, but practicality will limit the area
of search (note that a system starting
with 2¢, 5¢, 9¢ could not even do a
transaction for 1¢ using two coins).
2 A fruit-seller displays his oranges in square
boxes which take a whole number of
oranges on each side. The bottom layer
fills the box and higher layers are placed byEnd-of-chapter assignments