Thinking Skills: Critical Thinking and Problem Solving

3.6 Solving problems by searching 99

method involves analysing the problem,

which can be a very useful tool in reducing the

size of searches.

The type of search shown above involves

combining items in a systematic manner.

Other searches can involve route maps –

looking for the route that takes the shortest

time or covers the shortest distance, or

tables – for example finding the least

expensive way of posting a number of parcels.

With all these searches, the important

thing is to be systematic in carrying out the

search so that no possibilities are missed and

the method leads to the goal. The activity

below involves finding the shortest route for a

journey.

The map shows the roads between four

towns with distances in km.

I work in Picton and have to deliver

groceries to the other three towns in any

order, finally returning to Picton. What is the

minimum distance I have to drive?

14

Roseford

Queenstown

10

18

8

12

5

22

Southland

Picton

Activity

Commentary

There are only a small number of possible

routes. If these are laid out systematically and

the sums calculated correctly, the problem

can be solved quite quickly.

The possible routes (with no repeated visits

to any towns – you should be able to satisfy

combination is considered. The alternative
given in the above paragraph may be called a
‘directed search’ where we are looking
selectively for a solution and will give up the
search once we have found one. In the case
above, where we were looking for a minimum,
we can reasonably start searching from the
lowest value up.
A third alternative may be described as a
‘selective search’. In this case we are using a
partial analysis of the problem to reduce the
size of the search, concentrate on certain
areas, or to reject unlikely areas. The activity
below illustrates this.

Try repeating this exercise using coins of

denominations 1¢, 2¢, 5¢, 10¢, 20¢ and

50¢ and with 1 to 4 coins in each envelope.

This is quite a long search. Consider (and

discuss with others) whether there are ways

of shortening it.

Activity

Commentary
If you start this search you will find it takes a
very long time. It is difficult to be absolutely
systematic (especially when considering all
options for four coins). It is also difficult to
keep track of all values that have been covered
at any point in the search. It is necessary to
look for short-cuts, and out of boredom you
will probably have done so.
The denominations 1¢, 2¢ and 5¢ in
combinations of 1 to 3 coins can make all the
values from 1¢ to 10¢. This means that, by
adding to the 10¢ and 20¢ coins, all amounts
from 1¢ to 30¢ can be made. After 30¢ it is
necessary to use both the 10¢ and 20¢ or a 20¢
and 2 × 5¢. The former leaves one or two extra
coins, which can make 1¢, 2¢, 3¢, 4¢, 5¢, 6¢,
7¢ but not 8¢. The latter leaves only 1 coin,
which cannot be 8¢, so 38¢ is the minimum
that cannot be made from 1 to 4 coins. This