WRIGHT BROS NOT FIRST TO FLY
Unit 3 Problem solving: basic skills
Consider the action of making a cup of instant coffee. If you analyse the processes you need to go through, they are quite complicated. Just the list of items you need is quite long: a cup, a teaspoon, a jar of coffee, a kettle, water, and milk and sugar if you take them. Having found all these items, you fill the kettle and boil it;
use the teaspoon to put coffee into the cup;
pour the boiling water into the cup, just to the right level; stir; add milk and sugar; then put all the things you used away again. In fact one could break this down even more: we didn’t really go into very great detail on, for example, how you boil the kettle.
Although this is complicated, it is an everyday task that you do without thinking.
However, if you encounter something new, which may be no more complicated, the processes required to achieve the task may need considerable thought and planning.
Most of such planning is a matter of
proceeding in a logical manner, but it can also require mathematical tasks, often very simple, such as choosing which stamps to put on a letter. This thought and planning is what constitutes problem solving.
Solving most problems requires some sort of strategy – a method of proceeding from the beginning which may be systematic or may involve trial and error. This development of strategies is the heart of problem solving.
Imagine, for example, trying to fit a number of rectangular packages into a large box. There are two ways of starting. You can measure the large box and the small packages, and calculate the best way of fitting them in. You may make some initial assumptions about the
best orientation for the packages, which may turn out later to be wrong. Alternatively, you may do it by trial and error. If you have some left over at the end that are the wrong shape to fit into the spaces left, you may have to start again with a different arrangement. Either way, you will have to be systematic and need some sort of strategy.
With some problems the method of finding an answer might be quite clear. With others there may be no systematic method and you might have to use trial and error from the start.
Some will require a combination of both methods or can be solved in more than one way.
The words ‘problem solving’ are also used in a mathematical sense, where the solution sought is the proof of a proposition. ‘Problem- solving’ as tested in thinking skills
examinations does not ask for formal proofs, but rather asks for a solution, which may be a calculated value or a way of doing something.
Although many of the problems we shall look at here use numbers and require numerical solutions, the mathematics is usually very simple – much of it is normally learned in elementary education. Many problems do not use numbers at all.
As we saw in Chapter 1.3, there are three clearly defined processes that we may use when solving problems:
• identifying which pieces of data are relevant when faced with a mass of data, most of which is irrelevant
• combining pieces of information that may not appear to be related to give new information
• relating one set of information to another in a different form – this involves using experience: relating new problems to ones we have previously solved.
When solving problems, either in the real world or in examinations, you are given, or have, or can find, information in various forms – text, numbers, graphs or pictures – and need to use these to come up with a further piece of information which will be the solution to the problem.
The processes described above are the fundamental building blocks of problem- solving and can be expanded into areas of skill that may be brought together to solve more complex problems. The chapters in this unit divide these into smaller identifiable skill areas which can be tested using multiple- choice questions. Examples of such sub-skills are searching for solutions and spatial reasoning (dealing with shapes and patterns).
Later units deal with more complex problems, which can only be solved using several of these sub-skills in combination, and are closer to the sort of problem solving encountered in the real world.
The activity below gives an example of a simple problem; you can give either a simple answer or a more complicated one, depending on the degree of detail you consider necessary.
Luke has a meeting in a town 50 miles away at 3 p.m. tomorrow. He is planning to travel from the town where he lives to the town where the meeting is by train, walking to and from the station at both ends.
List the pieces of information Luke needs in order to decide what time he must leave home.
Then work out how you would proceed to plan his journey from these pieces of information.
Activity
Commentary
The chances are that you missed some vital things. You may have thought that all he needed was a railway timetable. Unless you approached the problem systematically, you may not have thought of everything.
Let us start by thinking of everything he does from leaving his house to arriving at the meeting.
1 He leaves his house.
2 He walks to the station.
3 He buys a train ticket.
4 He goes to the platform.
5 He boards the train when it arrives.
6 He sits on the train until it reaches the destination.
7 He leaves the train.
8 He walks to where his meeting is being held.
You can construct the pieces of information he needs from this list. They are:
1 The time taken to walk from his house to the station.
2 The time needed to buy a ticket.
(Remember to allow for queues!) 3 The time to walk to the platform.
4 The train timetable.
5 The time taken to walk from the station to where the meeting is being held.
Did you find them all? Perhaps you thought of some that I missed. For example, I didn’t think of allowing for the train being late. You could estimate this by experience and allow some extra time.
Now, to find out when he should leave home we need to work backwards. If his meeting is at 3 p.m., you can work out when he must leave the destination station to walk to the meeting.
You can then look at the timetable to see what is the latest train he can catch (allowing extra for the train to be late if appropriate). Then see from the timetable when this train leaves his home town. Continuing, you can determine when he should have bought his ticket, and when he should leave home.
3.1 What do we mean by a ‘problem’? 81 Of course, you could do the whole thing by
guesswork, but you might get it all wrong and, more to the point, you cannot be confident that you will have got it right.
In the sense we are using the word in this book, a ‘problem’ means a situation where we need to find a solution from a set of initial conditions. In the following chapters we shall look at different sorts of problem, different kinds of information, and how we can put them together to find solutions to the
problems. These chapters will lead you through the types of problem-solving exercises you will encounter in thinking skills examinations and give some indications about how you might
4 The following questions are based on a very simple situation, but require clear thinking to solve. Some are easier than others.
A drawer contains eight blue socks and eight black socks. It is dark and you cannot tell the difference between the two colours.
a What is the smallest number you will have to take out to ensure that you have a matching pair?
b What is the largest number you can take out and still not have a matching pair?
c What is the smallest number you can take out to be sure that you have one of each colour?
d What is the largest number you can take out and still have all of one colour?
e What is the smallest number you can take out to be sure you have a blue pair?
Answers and comments are on pages 315–16.
1 Imagine you are going to book tickets for a concert. List the pieces of information you need and the processes you need to go through in order to book the tickets and get to the concert. In what order should you do them? First list the main things, then try to break each down into smaller parts.
2 Consider something you might want to buy, such as a car, mobile phone or computer.
Make a list of the pieces of information you would need in order to make a decision on which make or model to buy.
3 Find a mileage chart that gives the distances between various towns (these can be found in most road atlases or on the internet). Pick a base town and four other towns. Consider making a journey that starts at the base town, takes in the other four and ends at the base town. In what order should you visit the towns to minimise the journey?
End-of-chapter assignments
approach such problems. However, learning to solve problems is a generally useful life skill and also, we hope, fun!
Summary
• In this chapter we have looked at what a problem is and how the word can be used in different ways.
• We have seen how information is used to contribute to the solution of a problem.
• We have looked at how various methods of using information can lead to effective solutions.
How do we solve problems?
3.2
We have seen that a problem consists of a set of information and a question to answer. In order to solve the problem we must use the information in a certain way. The way in which we use it may be quite straightforward – it may for example be simply a matter of searching a table for a piece of data that matches given conditions. In other cases, instead of searching for a piece of data, we may have to search for a method of solution. The important thing in either case will be to have a strategy that will lead to the solution.
Many publications give (in various forms) the procedure:
Data Process Solution This is all well and good, and indeed represents a way problems can be solved. It says nothing about what the words and, in particular, the arrows mean. It is in this detail that the key to problem solving is found. In simple terms, we are concerned with identifying the necessary pieces of data and finding a suitable process. There are no hard and fast rules; different problems must be approached in different ways. This is why problem solving appears in thinking skills examinations; it tests the ability of candidates to look at situations in different ways and to be able to use many different strategies to find one that works. Whilst a knowledge of the different categories of problem, as identified by the syllabuses and the various chapters of this unit, will help, you will always need to have an open mind and be prepared to try different approaches.
There are several ways problems may be approached. A term that is used a lot is
‘heuristic’ (see for example How to Solve It
by G. Polya [Penguin, 1990] – a book on mathematical problem solving). This word comes from the Greek ‘to find’ and refers to what we might call ‘trial and error’ methods.
Alternative methods depend on being systematic: for example, an exhaustive search may lead to an answer. Previous experience of solving similar types of questions will always be a help.
Imagine you are going out and can’t find your house keys. Finding them is a problem in the sense meant by this section of the book.
The heuristic method (and sometimes the quickest) is to run around all the likely places to see if they are there. After the likely places, you start looking at the less likely places, and so on until they turn up or you have to resort to more systematic methods. There are two systematic ways of searching. The first (using experience) involves thinking carefully about when you last came into the house and what you did; this can be the quickest method. The other (which in mathematical terms is often known as the ‘brute force’ method) involves searching every room of the house thoroughly until they are found. This is often the most reliable method but can take a very long time and most people will use it as a last resort.
When people are solving problems, they may use all of these methods, often in the order given above. This is quite logical, as the heuristic method can lead to a very rapid solution whilst the systematic search is slowest. One of the prime skills you need in tackling problem-solving questions in
examinations is to make a good judgement of which method is the most appropriate one to use in any set of circumstances.
3.2 How do we solve problems? 83 Commentary
The sum of the charges on the itemised bill is
$453. This is $18 less than her bill, so she has been overcharged for one dinner. None of the other items could come to exactly $18, either singly or severally.
Although this example is simple, it illustrates many of the methods used in solving problems:
• Identify clearly and unambiguously the solution that is required. Reading the question carefully and understanding it are very important.
• Look at the data provided. Identify which pieces are relevant and which are irrelevant.
• Do you need to make one or more intermediate calculations before you can reach the answer? This can define a strategy for solving the problem.
• You may need to search the given data for a piece of information that solves (or helps to solve) the problem.
• Past experience of similar problems helps.
If you had never seen this type of problem before, you would have had to spend more time understanding it.
• The above problem was solved using a systematic procedure (in this case calculating the correct bill, a value not given in the original problem).
The activity below, whilst still being relatively simple, involves a slightly different type of problem where the method of solution is less obvious.
The SuperSave supermarket sells Sudsy washing up liquid for $1.20 a bottle. At this price they are charging 50% more than the price at which they buy the item from the manufacturers. Next week SuperSave is
Activity
In any problem you will be presented with some initial pieces of information – these may be in the form of words, a table of numbers, a graph or a picture. You will also know what you need to produce as a
solution (the answer to a question). The first thing to do is to identify which pieces of information are most likely to be useful in proceeding to the solution and to try to work out how these pieces of information may be used. Problem-solving questions often contain redundant information, i.e. that which is not necessary to solve the problem.
This echoes real life, where the potential information is infinite.
The activity below is a relatively easy example. It is not difficult to find a way of approaching the problem, and the necessary calculations are clear and simple. See if you can do it (or at least work out how you would tackle it) before looking at the commentary which follows.
Julia has been staying in a hotel on a business trip. When she checks out, the hotel’s computer isn’t working, so the receptionist makes a bill by hand from the receipts, totalling $471. Julia thinks she has been overcharged, so she checks the itemised bill carefully.
Room: 4 nights at $76.00 per night Breakfast: 4 at $10.00 each
Dinners: 3 at $18.00 each
Telephone: 10 units at $1.70 per unit Bar: various drinks totalling $23.00 Laundry: 3 blouses at $5.00 each It appears that the receptionist miscounted one of the items when adding up the total. Which item has Julia been charged too much for?
Activity
having a ‘Buy two get a third free’ offer on this item. The supermarket does not want to lose money on this offer, so it expects the manufacturers to reduce their prices so SuperSave will make the same actual profit on every three bottles sold.
By how much will the manufacturers have to reduce their prices?
A 16 B 14 C 13 D 12 E 23
problems before and you think carefully about the information given.
Finally, to be sure that you have found the correct solution, check the answer. The profit on one bottle was $1.20 − 80¢ = 40¢; the profit on three bottles under the offer is $2.40 −
$1.20 = $1.20, or 40¢ per bottle. That’s correct!
You should have learned a little about finding a method of solution from this example. The guesswork method can only work by luck. This may be called the ‘pirate’s gold’ approach – we know the treasure is on the island somewhere so we dig a hole. If it’s not there, we dig another one somewhere else. Sometimes this method may seem to work, but it is usually because a little previous experience has been used, even unknowingly. The trial and error method, sometimes using a common sense strategy which turns it into a partial search, can be effective for solving some problems. Other problems may need an exhaustive search to solve; these are discussed in Chapter 3.6.
In the case above – and in many others – the method of finding a clear strategy was the most efficient. Strategies are not always found by rigorous methods; the discovery of an appropriate strategy usually depends on past experience of similar problems.
• We have looked at some methods of solving problems, investigating how different methods may be used in different circumstances.
• We have recognised the value of
experience in identifying problem types and appropriate methods of solution.
• We have seen how important it is to read and understand the information and the question.
• We have looked at the relative merits of guesswork, searching and strategic methods of solution.
Summary
Commentary
This could be solved in a variety of ways. We could just guess. As we are letting 13 of the bottles go for free, option C, 13, is tempting.
This is wrong.
It could be done by trial and error: for example, start with the manufacturers charging 60¢ (this would be option B) and see what that leads to. For three bottles they will charge $1.80 and the supermarket sells for
$2.40, so their mark-up is 60¢ for three bottles or 20¢ each. This is not enough, so the manufacturers’ price must be lower.
In fact there is a straightforward, systematic way of solving this which is made clear by writing down all the relevant values which can be calculated:
Normally SuperSave sell at $1.20, so they buy at 80¢ (selling at 50% more than they buy), so each bottle is sold for 40¢ more than the price at which it is bought.
Under the offer, they will sell three bottles for the price of two, i.e. three for $2.40, or 80c each. If they are still selling for 40¢ more than the price at which they have to buy, they will be buying from the manufacturer at 40¢. So, the manufacturers will have to halve their price. Option D is correct.
This method was quite quick, and certainly quicker than the trial and error method. It is the sort of solution that you are more likely to come up with if you have seen a lot of similar