In psychology a problem is defined as a situation in which some of the components are already known and additional components must then be ascertained or determined. Such situations are of interest to psychologists when these unknown factors are neither obvious, nor easily ascertained.
Problem solving is broadly all the processes involved in the solution of a problem, and in psychology is the area of cognitive psychology that is concerned with these processes. Examples of classic puzzles used by psychologists as exercises in problem solving are anagrams, puzzle boxes and water-jug problems.
The following is a typical example of the type of puzzle which concerns itself with the process of problem solving:
How is it possible to measure 9 minutes when you have only a 4-minute sand glass and a 7-minute sand glass?
Start both glasses running at the same time. Turn the 4-minute glass over as soon as it runs out (4 minutes). Then turn the 7-minute glass over when it runs out (an extra 3 minutes). At that moment, the 4-minute glass has 1 minute left. Turn it over when that 1 minute is over (plus 1 minute). Then turn the 7-minute glass over, which ran for only 1 minute (plus 1 minute). That makes a total of 9 minutes (4 + 3 + 1 + 1).
Of course, real-life problems are a somewhat different proposition but the thought processes necessary in tackling them are the same as those involved in the analysis, and finding of solutions, of problems, or puzzles, that are artificially set.
The subde difference between puzzles and problems is that a puzzle is set by another person, and has a solution already known to that person. A problem, on the other hand, arises in life. It is not set artificially and there is not an answer already known to someone else. There is no right answer but some solutions are likely to be better than others.
One of the key points to bear in mind when tackling a real-life problem is not to spend too much time complaining about the problem or falling into the trap of wallowing in self-pity Instead of this it is much better to talk at the problem and not about it. Admittedly, for many serious personal problems this is much easier said than done, but in almost all cases a positive approach is much better than a negative one.
With many problems or difficult situations it is essential to get to the heart of the problem. In one of the opening lines of the 1946 movie of Raymond Chandler's The Big Sleep, Humphrey Bogart, as Philip Marlowe, remarks, 'Get Mr Big and you get them all.' This is true of many problems, where if you get right to the heart of the problem and can find a solution to the main cause, then it is amazing how quickly all the peripheral problems also disappear.
There are many different problem-solving techniques that have been put into practice over the years. Perhaps the best known of these is brainstorming which is a group problem-solving technique in which all the participants are encouraged to let fly with ideas and possible solutions to the problem in hand.
Brainstorming is, therefore, a method of searching for, and developing creative solutions to a problem by focusing on the problem and deliberately encouraging the participants to come up with as many unusual solutions as possible. During such sessions there should be no criticism of ideas as the aim is to introduce as many different ideas as possible, and to break down any preconceptions about limits of the problem. Then, once this has been done, the results and ideas can be analysed and the best proposed solutions further explored.
Participants in such brainstorming sessions need not necessarily be experts in the field under scrutiny, nor should they necessarily be already aware of the problem under consideration. They should ideally come from as wide a range of disciplines and backgrounds as possible. This brings many more creative ideas to the session, and often someone looking at a problem from the outside may suddenly come up with a possible solution that someone heavily involved on the inside has not considered.
Although it is generally not as effective as a group session, brainstorming can also be carried out individually. While such individual brainstorming can sometimes produce a wider range of ideas than a group session, these ideas tend not to be developed as effectively because individually it is more difficult to tackle and solve any additional problems that may be encountered. On the other hand, individuals are free to explore their ideas in their own time without being pressurized by other group members and without fear of criticism.
One of the great benefits of individual brainstorming is that it is a method of attacking a problem and encouraging your brain to function in a creative and positive way by exploring new ideas and solutions.
Another tried and tested method of problem solving is Critical Path Analysis. As a mind tool this is an effective method of analysing a complex problem or project, and it is a particularly useful tool where there is a time factor or deadline involved.
The concept behind critical path analysis is the need to formulate a plan of action because some activities are dependent on other activities being completed first. For example, you cannot convert your garage into a living room unless you have drawn up plans, obtained pricing estimates and, in some cases, obtained planning permission. Such dependent activities need, therefore, to be completed in sequence, with each activity usually completed, or nearly completed, before the next activity can begin.
A third problem-solving technique is known as SWOT Analysis - Strengths, Weaknesses, Opportunities, Threats. This technique can be an effective method of identifying your strengths and weaknesses and of examining the opportunities and threats faced.
In order to carry out a SWOT analysis, start by writing down answers to the following questions:
What are your strengths?
What are you good at?
These questions should be considered both from your own point of view and from the point of view of others you deal with. At all times you must be honest and realistic with yourself and able to recognize many of your own characteristics, many of which could turn out to be your strengths.
In what ways can you improve? What do you do badly? What should you avoid doing?
Again, you must be realistic for the exercise to be effective, and again you should answer not just from your own point of view, but from the point of view of others.
What opportunities, immediate or otherwise, would be good for you? What would be most interesting to you?
Anything relevant, or possibly relevant should be considered, for example, changes in technology, lifestyle changes, career advancement.
What obstacles do you face? What financial constraints exist? What specifications must I adhere to?
Whether the problem, or task, is a small or a large one, the carrying out of such an analysis can be an illuminating one, not just in identifying what needs to be done, but in putting any problems into perspective and helping to identify your own strengths, which can then be built upon, and your own weaknesses, which can then be worked upon.
In the remainder of this chapter we present a collection of 25 general brainteasers which involve different kinds of thought processes, followed by a collection of 20 numerical brainteasers, also involving a variety of different types of approaches. For the more difficult puzzles, hints are provided and full detailed explanations are provided with the answers.
While both puzzles and problems bring their rewards, some people may prefer one to the other. Certainly, the successful solution of a problem achieves a worthwhile goal, and perhaps the major benefit to be obtained from tackling puzzles is that they stretch and exercise the mind and enable you to tackle the problems of real life with renewed vigour and confidence.
Before tackling the following selection of puzzles it may help you to consider the following four brainteasers as examples of the thought processes necessary to solve such puzzles.
There are several versions of counterfeit coin puzzles where it is necessary to find the counterfeit coin, or coins, from a number of weighings. Most of these puzzles assume you have balance type scales available with two pans, where one object can be weighed against another. In this puzzle you have a single scale only with just one pan, which will weigh just one object, or group of objects at a time. You have three bags with an unspecified number of coins in each bag. One of the bags consists entirely of counterfeit coins weighing 75 grams each; the genuine coins weigh 70 grams each.
What is the minimum number of weighing operations you need to carry out before you can be certain of identifying the bag of counterfeit coins?
Only one weighing operation.
Take one coin from bag 1, two coins from bag 2 and three coins from bag 3. Weigh all six coins together. If they weigh 425g, the first bag contains the counterfeit coin (1 coin at 75g + 5 coins at 70g = 425), if they weigh 430g, it is the second bag, and if they weigh 435g, it is the third bag.
A company gives a choice of two plans to the union negotiator for an annual increase in salary.
First option: Initial salary £20000 to be increased after each 12 months by £500.
Second option: Initial salary £20000 to be increased after each six months by £125.
The salary is to be calculated every 6 months. Which plan should the union negotiator recommend to his members?
The second plan.
At first glance it appears obvious that the union negotiator should recommend the first option as this will give £500 increase per year and the second will only give £250 increase per year. However, this is not the case. Let us examine the two options more closely:
First option (£500 increase after each 12 months):
First year £10000 + £10000 = £20000 Second year £10250 + £10250 = £20500
Second option (£125 increase after each 6 months):
First year £10000 + £10125 = £20125 Second year £10250 + £10375 = £20625
Two identical bags each contain 8 counters, 4 white and 4 black. One counter is drawn out of bag one and another counter out of bag two. What are the chances that at least one of the balls will be black?
Three chances in four. Explanation:
Look at the possible combinations of drawing a counter from each bag. These are:
a. black - black b. white - white c. black - white d. white - black
Out of these four possible combinations there is only one, the white - white combination, where black does not occur. The chances, therefore, of drawing at least one black ball are three chances in four.
Ail five circles have the same diameter. Draw a line passing through point A in such a way that it divides the five circles into two equal areas.
By drawing three additional imaginary circles to create a symmetrical block of eight circles, the problem is considerably simplified and leads to the following solution:
The puzzles (Answers, see pp. 185-92)
A man walks through a park from West to East but decides he needs to take some extra exercise by walking around every one of the park's three circular paths instead of just walking along the main central drive, for example, the dotted line shows one particular route which he can take to achieve this objective. However, how many different routes are there by which he is able to do this? He does not go over any part of the route twice but does, inevitably, arrive at the same point more than once on his travels.
2 In the game of draughts a piece (draughtsman) can jump over another piece in any direction, including diagonally. A draughtsman is removed when another piece jumps over it. In one move you can make your piece jump over a series of other pieces.
Make just one move so that only eight pieces remain on the board, so arranged that no horizontal, vertical or corner to corner line contains more than one piece.
What is the ratio between the two shaded areas in the diagram above? (Hint, see p. 160)
Without looking at a real die, can you quickly calculate the total of spots on faces 1, 2, 3, 4, 5, 6 and 7? (Hint, see p. 160)
5 I want to hold a dinner party,' said my wife, but I want to invite only a very few guests.'
'OK,' I said, 'then as well as the two of us, I suggest you invite your sister Jane, her brother-in-law and his wife, my son Keith, my book editor Alex and her husband and son, and Mrs Morgan, the widow from next door, and her nephew who I happen to know is visiting her tomorrow.'
'That sounds to be more than I was intending to invite,' said my wife. 'No it isn't,' was my reply, 'It is actually very few people, just think about it.'
How many were at the dinner party? (Hint, see p. 160)
6 A ball is put in an empty bag. You do not know whether the ball is black or white. A second ball which you know to be white is then put in the bag. A ball is then drawn out, and it proves to be white.
What are the chances that the ball remaining in the bag is also white?
7 I was on the third floor of a department store recently with my wife and we decided to take the lift to the top floor. When we pushed the button the lift was already on the 7th floor, after which it went up to the 9th floor, down to the 6th, back up to the 11th and down to the 4th.
'This is hopeless,' I said, 'we may as well walk.' 'No,' said my wife, 'just be patient, the lift will now go up to the 12 th floor, then it will come back to us at last, however, it will then take us up to a certain floor whether we like it or not.'
How did my wife know that it would visit us after the 12th floor, and which floor would it then take us to, whether we liked it or not? (Hint, see p. 161)
Which is the missing square?
9 68932, 71456, 98372, 14568 What comes next?
56381, 89372, 29347, 82943 or 75286? (Hint, see p. 161)
10 Heathcliffe sent Cathy the cryptic message below. What does it mean?
Cathy footworn abyss wither
Arrange the digits 1-9 in the circles in such a way that.
Numbers 1 and 2 and all the digits between them add up to 9.
Numbers 2 and 3 and all the digits between them add up to 19.
Numbers 3 and 4 and all the digits between them add up to 45.
Numbers 4 and 5 and all the digits between them add up to 18.
13 In a bag of apples, 4 out of 50 contain a worm. What are the chances of picking out just 3 apples and finding they all contain a worm? (Hint, see p. 161)
14 Can you draw the missing arrangement of dots?
Which is the missing square? (Hint, see p. 161)
16 How many triangles occur in the figure below?
The game of pool or snooker calls for a high degree of skill and concentration, coupled with a great deal of creative thinking. In this example you have just one ball, the black ball, on the table and you need to pocket it with the white ball. Several of your opponent's balls, the striped balls, are on the table and you must figure out how to shoot your last remaining ball into a pocket without touching any of the striped balls in the process. Work out how you can strike the white ball to travel round the table and knock the black ball into a pocket by rebounding off the minimum number of sides of the table.
18 I did the City of Leeds half-marathon recently. However, after I had completed the first two-thirds of the course I developed a blister on my foot and had to hobble the rest of the way to the finishing line. It took me just twice as long to hobble the remainder of the race as it did to run the first two thirds.
How many times faster did I run than I hobbled?
19 What number should replace the question mark?
20 point + pivot = cavity advice + drivel = taxes knives + knife = sever Therefore:
Bahrain, China, Brazil, Mexico, Latvia (Hint, see p. 162)
21 If we presented you with the words MAR, AM and FAR and asked you to find the smallest word that contained all the letters from which these words could be produced, we would expect you to come up with the word FARM.
Here is a further list of words:
DAMSEL, LEASH, MELODY, LEMON
What is the shortest English word from which all these four words can be produced? (Hint, see p. 162)
22 Which group of letters is the odd one out?
LNQP JLNM DFIH GILK SUXW
23 Paul has 26 cards each featuring a different letter of the alphabet which he places face down on the table and then turns over at random one by one.
What are the odds that the first four cards to be turned over will spell out his name P_A_U_L in the correct order, and what are the chances that the last four cards, instead of the first four cards, will spell out P_A_U_L in the correct order? (Hint, see p. 162)
24 What message is concealed in the array of words below?
25 What arrangements of dots should replace the question marks? (Hint, see p. 162)
Numerical problem solving
Mathematics may be defined as the subject we never know what we are talking about, nor whether what we are saying is true.
Problems involving the use of mathematics can be challenging, fascinating, confusing and frustrating, but once you have developed an interest in numbers, a whole new world is opened up as you discover their many characteristics and patterns.
We all require some numerical skills in our lives, whether it is to calculate our weekly shopping bill or to budget how to use our monthly income, however, for many people mathematics is a subject they consider to be too difficult when confronted by what are considered to be its higher branches. When broken down and analysed, and explained in layman's terms, however, many of these aspects can be readily understood by those of us with only a rudimentary grasp of the subject.
The following section consists of a set of 20 difficult numerical problems which involve several kinds of calculation, reasoning and logic. The more you practise on this type of puzzle, the more you will come to understand the thought processes and analysis necessary to solve them and the more proficient you will become at arriving at the correct solution.
Hints are provided for the most difficult puzzles and in all cases we provide full detailed explanations as to how you should tackle the problem to arrive at the correct solution.
1 If a half of 5 were 3, what would one-third of 10 be?
2 Jimmy has 10 pockets and 44 coins. Each coin value = 1 Euro.
He wants to place the 44 coins into his 10 pockets so that each pocket contains a different number of coins.
Can he do this?
3 At a college 100 students were studying languages.
27 students studied Latin
49 students studied French
35 students studied German
8 students studied Latin/French 6 students studied Latin/German
9 students studied French/German
3 students studied Latin/French/German
How many students studied none of the 3 languages? (Hints for puzzles 1, 2, 3, see p. 163)
4 Two golfers, Geoff and Harry, decided to have a wager on the golf course. They intended to play 18 holes and have a wager on each hole. Geoff said to Harry, 'We will wager on each hole, half of the money in my wallet. I have £100 in my wallet.'
After playing 12 holes it began to rain so they retired to the club house. As Geoff had won 6 holes, Harry had won 4 holes and 2 holes were tied, Geoff said, 'I wili buy the drinks.' However, looking in his wallet he found that he had lost £28. How did that happen?
5 In a room that is full of aliens from another planet:
1 There is more than one alien.
2 Each alien has the same number of fingers.
3 Each alien has at least one finger on each of his hands.
4 In the room the total number of fingers lies between 200 and 300.
5 If you knew the total number of fingers in the room, you would know how many aliens there were.
How many aliens were there?
How many fingers did each alien have?
I wanted to know the month when the circus was coming to town, so I asked six of my friends. These were their answers:
Alan said: 'It begins with the letter J.' Barbara said: 'It has only 5 letters in its name.' Carol said: 'It has 30 days in its month.' David said: 'It has 31 days in its month.' Edward said: 'It has 3 vowels in its month.' Fiona said: 'It ends in Y'
But half had lied.
Which month was it?
7 A bookmaker was lying odds on a race:
SANDS OF TIME LITTLE BEN MOONSHINE PHIL'S FOLLY ABANDON ALL HOPE CATCH KEN
4-1 Against 8-5 Against
10-1 Against ?
CATCH KEN has no odds yet decided. What odds should the bookmaker offer to give himself a margin on 15%, assuming that he is able to balance his books?
8 An old fairground game consisted of a sheet of linoleum which had a pattern of 4" squares drawn on it.
If the player threw a 2V2" diameter coin on it, what are the chances that the coin will fall not touching a line? (Hint, see p. 164)
9 What number should replace the question mark? (Hint, see p. 165)
10 How many revolutions must the large cog make to return all the cogs to their starting positions? (Hint, see p. 165)
9 What number should replace the question mark? (Hint, see p. 165)
10 How many revolutions must the large cog make to return all the cogs to their starting positions? (Hint, see p. 165)
11 When the Rajah died he left a box containing a collection of diamonds.
To his eldest son he left one diamond, to the eldest son's wife he left one-ninth of those remaining.
To his second son he left 2 diamonds, to the second son's wife he left one-ninth of those remaining.
To his third son he left 3 diamonds, to the third son's wife he left one ninth of those remaining.
And so on.
The wife of the youngest son found that there were no diamonds left for her.
How many diamonds did the Rajah have, and how many sons did he have?
12 There is a bamboo cane 10 feet high, the upper end of which has been broken down and now reaches the ground, its tip lying just 3 feet from the stem.
Starting at the top right-hand corner and spiralling inwards, how far would you have to travel to arrive at the centre? (Hint, see p. 166)
14 The state of Cattattackya was overrun with mice, so the King decreed that all the cats should exterminate the vermin.
At the end of the year the dead mice were counted and the total was 1,111,111, each cat had killed an equal number of mice.
Less than 500 cats achieved this remarkable feat. How many cats were there in Cattattackya?
15 Jim is trying to find out where Tim lives. He knows that the numbers in his road are 8 to 100 inclusive.
Jim asks, 'Is it greater than 50?' and Tim answers and lies.
Then Jim asks, 'Is the number a multiple of 4?' and Tim answers and again lies.
Then Jim asks, 'Is it a perfect square?' and Tim answers and this time he tells the truth.
Then Jim asks, 'Is the first digit 3?' and Tim replies (truthfully or not).
Jim tells him his number but he is wrong. What is the number?
16 A woman had seven children. On multiplying their ages together one obtains the number 6591.
Given that today is the birthday of all seven, how many triplets are there and what are all seven ages?
17 /*> is irrational. Its decimal goes on to infinity. To 6 decimal places it is 2.236068.
Find the value of:
3 by a simple calculation.
Find which digit replaces the question mark.
19 What are the true odds against winning the lottery of 49 numbers?
20 The carpet in a room was coloured as follows: V3 is black, V4 is red, and the remainder is yellow measuring 8 square yards.
What is the total area of the room?
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