3.6
Some problems may not always be resolved by using direct methods of calculation.
Sometimes, problems do not have a single solution, but many, and we need to find one that represents a maximum or minimum (for example the least cost or shortest time for a journey). In these cases we need to have a systematic method of evaluating the data to come up with all (or at least the most likely) possibilities. This is called a ‘search’. Once again, with this type of question, it is
important to have a way of checking that the final answer is correct.
Here is an example of a problem that requires a search.
Amir is helping with a charity collection and has gathered envelopes containing coins from a number of donors. He notes that all the envelopes contain exactly three items but some of them contain one, two or three buttons instead of coins. All the coins have denominations of 1¢, 5¢, 10¢, 25¢ or 50¢.
What is the smallest amount of money that is not possible in one of the envelopes?
Activity
Commentary
The easiest way to approach this question is to list the possibilities in a systematic order. We know envelopes can contain 0, 1, 2 or 3 coins.
The possibilities with one coin (and two buttons) are: 1¢, 5¢, 10¢, 25¢ or 50¢.
That was the easy part. With two coins (and one button), we need to be a little more careful. First consider that the first coin is 1¢, then look at all the possibilities for the second.
We can then continue with the first coin as a 5¢ in the same manner (we do not need to consider repeats). The possibilities are:
1¢ + 1¢, 1¢ + 5¢, 1¢ + 10¢, 1¢ + 25¢, 1¢ + 50¢, then
5¢ + 5¢, 5¢ + 10¢, 5¢ + 25¢, 5¢ + 50¢, 10¢ + 10¢, 10¢ + 25¢, 10¢ + 50¢, 25¢ + 25¢, 25¢ + 50¢, and 50¢ + 50¢.
Listing all the totals, we have: 2¢, 6¢, 11¢, 26¢, 51¢, 10¢, 15¢, 30¢, 55¢, 20¢, 35¢, 60¢, 50¢, 75¢ and $1.
Finally, we need to list all the possibilities with three coins. This is slightly more difficult.
However, we only need to go on until we have found an impossible amount (you may already have spotted it). The possibilities are:
1¢ + 1¢ + 1¢, 1¢ + 1¢ + 5¢ etc.
1¢ + 5¢ + 5¢ etc.
You should have spotted by now that we have not seen the value 4¢ and that all further sums of three coins (anything including a 5¢ or above) will be more than 4¢. So 4¢ is the answer.
This was actually a trivial example used for the purposes of illustration. There is an alternative way to solve this, which also involves a search. This is to look at 1¢, 2¢, 3¢, etc. and see whether we can make the amount up from one, two or three coins. In this case it would have led to a very fast solution, but if the first impossible value had been, for
example, 41¢, this second method would have taken a very long time and we might have been unsure that we checked every possible sum carefully.
The method described above is called an
‘exhaustive search’, where every possible
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
Therefore, the minimum distance is 59 km.
If you were particularly astute, you would have noticed that the routes come in three pairs of the same distance (e.g. PQRSP is the reverse of PSRQP so must be the same).
This would have saved you half the calculations.
Summary
• We have learned that some problems require a search to produce a solution.
• We have seen the importance of being systematic with a search, in order both to ensure that the correct answer is obtained and to be certain that we have the right answer.
• We also saw that searches do not always have to be exhaustive and how analysis of the problem can reduce the size of the search and time taken.
yourself that it is never worth retracing your steps) are:
PQRSP PQSRP PRQSP PRSQP PSRQP PSQRP
This gives six values. In order to see how they are obtained you may note that there are three pairs, each pair visiting a different one of the towns first (after leaving Picton). The two routes in each pair take the last two towns in opposite orders.
The distances associated with each route are as follows:
59 km 62 km 67 km 62 km 59 km 67 km
Additional adult $10
Family ticket (for 1 adult and 2
children) $20
Additional child 4–16 or senior
citizen $5
Additional adult $10
Maria is taking her three children aged 3, 7 and 10 and two friends of the older children (of the same ages) as well as her mother, who is a pensioner. What is the least it will cost them?
1 The notice below shows admission prices to the Tooney Tracks theme park.
Adult $12
Child (aged 4–16) $6
Child (aged under 4) Free
Senior citizen $8
Family ticket (for 2 adults and
2 children) $30
Additional child 4–16 or senior
citizen $5
End-of-chapter assignments
If 5¢ and 20¢ coins are the same thickness, how many different heights of
$1 pile could she have?
A 5 B 6 C 10 D 11 E 20
4 In a community centre quiz evening, teams were awarded five points for a correct answer, no points for no answer, and minus two points for an incorrect answer. The teams marked their own score sheets. I arrived late and the scores after seven questions were shown on the board as follows:
Happy Hunters 28
Ignorant Idlers 18
Jumping Jacks 16
Kool Kats 12
Lazy Lurkers -1
a One team was clearly not even clever enough to calculate their score correctly.
Which one was it?
b Are there any scores, other than those shown above, that would have raised suspicion?
Answers and comments are on pages 318–19.
2 I recently received a catalogue from a book club. I want to order seven books from their list. However, I noticed that their price structure for postage was very strange:
Number of
items Cost of post and packing
1 45¢
2 65¢
3 90¢
4 $1.20
5 $1.50
6 or more $3.20
I decide, on the basis of this, that I will ask them to pack my order in the number of parcels that will attract the lowest post and packing charge. How much will I have to pay?
3 Jasmine has been saving all year for her brother’s birthday. She has collected all the 5¢ and 20¢ coins she had from her change in her piggy bank. She is now counting the money by putting it into piles, all containing $1 worth of coins. She notices that she has a number of piles of different heights.
An extension of this skill is to identify possible reasons for variation in data – once again, this springs from past experience as to what causes changes and the types of variation that may be expected. This type of question is dealt with in more detail in Chapter 3.8.
These are best illustrated using examples.
The first deals with identifying the similarity between two sets of data.