The document discusses problem-solving strategies and logical reasoning, providing examples such as using jugs to measure water, interrogating suspects in a robbery, and navigating doors with either treasure or danger. It highlights stages of problem-solving, including defining the problem, lateral thinking, and various strategies like exhaustive search and abstraction. Additionally, it covers methods for simplifying problems, such as decomposition and functional abstraction, to create effective solutions.

1. Problem Solving
Logical reasoning or deductive reasoning.
Using a set of facts to determine whether a new set of facts are true or
false.
Some problems to consider:
1) There are two jugs. A and B. Jug A has a capacity of three litres. Jug
B has a capacity of five litres. There are no markings on the jugs, so
it is not possible to tell exactly how much is in a jug just by looking
at it, unless it is full or empty. There is a sink with a water tap and a
drain. How can exactly 4 litres of water be obtained from the tap
using the jugs
Step Jug A (3) Jug B (5)
1

2. Problem Solving
Some problems to consider:
Step Jug A (3) Jug B (5)
1 3 0
2 0 3
3 3 3
4 1 5
5 1 0
6 0 1
7 3 1
8 0 4

3. Problem Solving
Some problems to consider:
2) The police are interrogating five suspects after a bank robbery. Each
of them makes two statements, but it turns out only five of these
statements are true.
Can you work out who committed the crime? What strategy will you
use to work it out?
Alan said: It wasn’t Ben. It was Carl.
Ben said: It wasn’t Alan. It was Dave.
Carl said: It wasn’t Dave. It wasn’t Ben, either.
Dave said: It wasn’t Eddie. It was Alan.
Eddie said: It wasn’t Carl. It was Derek.

4. Problem Solving
Some problems to consider:
3)You are in a room with 2 doors leading out. Behind 1 door is a coffer
overflowing with jewels and gold, along with an exit. Behind the other
door is an enormous, hungry lion that will pounce on anyone opening
the door.You do not know which door leads to the treasure and exit, and
which door leads to the lion. In the room you are in are 2 individuals.
The first is a knight, who always tells the truth, and a knave, who
always lies. Both of these individuals know what is behind each door.
You do not know which individual is the knight, or which one is the
knave.You may ask one of the individuals exactly 1 question.What
should you ask in order to be certain that you will open the door with
the coffer behind it, instead of the hungry lion?

5. Stages of Problem Solving
 Defining the problem.
 A well defined problem has four components
1. A clearly defined initial condition given.
2. A clearly defined goal.
3. A clearly defined set of resources and constraints.
4. Ownership.
For example:
You are at a river that you want to cross with all your goods. Your goods consist
of a chicken, a bag of grain and your dog, Rover. You have to cross the river in
your rowing boat but can only take one passenger at a time – the chicken, the
dog or the bag of grain. You can’t leave the chicken alone with the grain as the
chicken will eat the grain. You can’t leave your dog alone with the chicken as
Rover will eat the chicken. However you know that Rover does not eat grain.
2

6. Stages of Problem Solving
So lets break this problem down into its components
Initial Situation
You, the chicken, the bag of grain and the dog are on one bank of a river with
access to a rowing boat.
Resources
The rowing boat and your knowledge and problem solving skill,
Constraints
You can only take one passenger, you must not leave Rover with the chicken or
the chicken with the grain.
Goal
You, the chicken, the bag of grain and the dog on the opposite bank of river.
Ownership
You will be involved in planning the solution and carrying it out.
CANYOU COME UPWITH A SOLUTION? 2

7. Stages of Problem Solving
Lateral Thinking.
Some problems don’t give you all the facts and you have to
Use lateral thinking to establish some more.
Puzzle:
Six glasses are arranged in a line. The first three are filled with water, the
second three are empty. You are allowed to move only a single glass. In one
move can you achieve an ordering of the glasses which alternates full and
empty glasses along the line?
• What constraint or boundaries did you assume?
Puzzle:
A man lives on the 18th
floor of a tower block. Every weekday morning he
takes the lift to the ground floor and walks to work. Every weekday evening, he
walks from work, takes the lift to the 12th
floor and then takes the stairs to the
18th
floor. If it was raining in the morning, he takes the lift straight to the 18th
floor in the evening. Why?
In all problems challenge your assumptions! 2

8. Problem Solving
Strategies for problem solving
1) Exhaustive search
You are to fill a square with 9 numbers. Each Row, column and diagonal
needs to add up to the same value.
This is a specific instance (0-9 & 3X3) of a general problem (N-N & NXN)
How would you solve this problem?

9. Problem Solving
Strategies for problem solving
1) Exhaustive search
One method is to use an EXHAUSTIVE SEARCH = which simply means
trying ever possible combination until you get it.
This would work out as 9*8*7*6*5*4*3*2*1 which = 362,880 possible
ways of filling the square or 9!
Size of the problem – 5X5 = 25! = 1.55 * 1025
possible ways of filling the
square.

10. Problem Solving
Strategies for problem solving
1) Exhaustive search
Needs a better algorithm
What if we knew that the middle number has to be 5 and each 3
numbers had to add up to 15 – would be easier.

11. Problem Solving
Strategies for problem solving
2) Divide and conquer or binary search
Ask a friend to think of a number between 1 and 1000. You have a guess
and your friend tells you whether it is correct, too low or too high. What
is the maximum number of guesses you will need with this form of
search method?
How many guess between 1 and 2N
How many guesses would this take using an exhaustive search?

12. Problem Solving
Strategies for problem solving
3) A flash of insight.
You have a bag of 8 coins which all look identical, but one of them is
counterfeit and weighs less than a genuine coin. You have a two-pan
balance, but no weights.
What is the maximum number of weighing's needed to identify the fake
coin?
The trick is to think of a method that reduces the need of using an
exhaustive search.
This is where developing good algorithms

13. Abstraction
• The process of reducing a problem to their essential features.
• Finding similarities or common aspects about a problem and
ignoring the differences.
• View from a high level – ignoring the detail.
We use abstractions every day

14. Abstraction
• Two types of abstraction
• Representational Abstraction
• Generalisation/Categorisation

15. Abstraction – Representational
• Process of removing unnecessary details until it is possible to
represent the problem in a way that can be solved.
• Looking more at the big picture.
• See previous slide with the London Underground
• Useful for someone travelling around London.
• Is it useful for an engineer wanting to build a new Line in the
underground?
• Sat Nav Example:
 Identify point A and point B in someway
 Identify the connecting paths between A and B
 Calculate the shortest path between A and B.
 Remove irrelevant information such as rivers, railways,
landmarks etc.

16. Abstraction –Generalisation / Catergorisation
• Placing aspects of problem into broader categories to arrive at a
Hierarchical representation
• Used in Object orientated programming.
Vehicles
Car Bus Lorry Van

17. Information Hiding
• Ties in with abstraction – The removing or all irrelevant details from
a problem.
• For example – the Bridge problem. 200 years ago in the old Prussian
city of Koingsberg. It had seven bridges and the inhabitants likes to
stroll around the city on a Sunday afternoon, making sure to cross
every bridge at least once. Nobody could figure out how to cross
each bridge once and once only. Leonhard Eular used Information
hiding to solve this problem.

18. Information Hiding
• The map of the city with the bridges.

19. Information Hiding
• Remove all irrelevant details from the map to come up with an
abstraction

20. Information Hiding
• Create a graph with vertices and edges to represent the bridges and
islands
• Euler concluded that each node needs to have an even number of
connections to enter and leave in order for each bridge to be crossed
once – therefore the above is impossible.
• Set a foundation to solve similar problems.
North
Bank
South
Bank
West
Island
East
Island

21. Abstraction – Procedural Abstraction
• The concept that all solutions can be broken down into a series of sub
problems.
• How programs written in code work.
• Design stage – sufficient to state what each procedure does not how it
does it. Values are hidden from each procedure.
• Amendments can be isolated and changes made easily without
affecting the whole solution.
Procedure for
Drawing
Street
Procedure for
drawing
house
Procedures
for drawing
shapes.
Hides detail from
Hides detail from Depends on
Depends on

22. Abstraction – Functional Abstraction
• Similar to Procedural Abstraction
• The use of common functions to solve problems – generic across
many problems.
• Also can be the actual application of the procedures mentioned last
slide - in a coding language.
• The methods and data of the function are hidden from the user.
e.g. Console.WriteLine()
Console.ReadLine()

23. Abstraction – Data Abstraction
• Process of organising and structuring data in a way that is useful to
the programmer but how it is actually represented in the computer is
hidden.
• Data types such as integers and strings have some abstraction to
them.
• Higher level data abstractions such as queues, stacks, linked lists,
trees etc.
• The hiding away of their representation allows you to build new data
types fairly easy – such as stacks from arrays – or records from a
number of data types.
• A separation of the actual implementation from the interface or use.

24. Abstraction – Problem Abstraction
• Removing details until the problem is represented in a way that is
possible to solve.
How many moves
would it take to
swap the knights
around.

25. 1 2 3
4 5 6
7 8 9
Abstraction – Problem Abstraction
• Number the board
How many moves
would it take to
swap the knights
around.

26. Abstraction – Problem Abstraction
• Create a graph joining up the numbers that are connected. 5 Has
been removed.
How many moves
would it take to
swap the knights
around.
1 3
2
4 6
9
7
8

27. Abstraction – Problem Abstraction
• Create a graph joining up the numbers that are connected. 5 Has
been removed.
Now it is a bit
easier to
visualise a
solution to this
problem.
6 8
1
7 3
4
2
9

28. Decomposition / Composition
• Breaking down large complex tasks or processes down into smaller
more manageable tasks.
• Abstraction techniques are applied in this process.
• Procedural Composition – Process of creating a working system from
the abstraction:
• Writing all the procedures and linking them together to create
compound procedures
• Creating data structures and combining them to form compound
structures.

29. Decomposition / Composition
• A Sat Nav System
Sat Nav
System
Journey
Start Point
End Point
Travel
Input travel
data
Input travel
updates
Road
Network
National
Roads
International
Roads
Calculate
Route

30. Automation
• Process of creating computer models of real-life situations and
putting them into action.
• Creates elegant solutions to difficult problems
• Understand the problem
• Begin able to create suitable algorithms
• Building up the algorithms into program code
• Using appropriate data in order to solve the problem.
Example – Traffic light system
• A specification needs to consider:
• Location of all traffic lights
• The number of roads that meet at each set of traffic lights
• How many lanes of traffic are at each set of lights
• How much traffic there is on each of the lanes
• What time of day it is.
• Do they control a pedestrian crossing as well.
Reviews of the system in play may required the rewriting of algorithms