Simple Deductive Logic Problems
Ho Soo Thong
Copyright © Oct 2015 AceMath, Singapore
This article aims to illustrate important pictorial approach to
some deductive logic problems at Primary Olympiad level.
1. Cubes with marked faces
We begin with a simple problem which requires making deduction by rules on marked faces of cubes.
Example 1
The six faces of a cube were labelled with letters A, B, C, D, E and F.
Figure 1 shows views from three different angles.
What is the letter opposite of A?
What is the letter opposite of B?
Show the net diagram of the die.
F
B
D
A
C
E
A
D
C
Figure 1
Solution
Any labelled face has 4 adjacent faces and one opposite face.
In Figure 2, the table lists the adjacent faces of A and C (both
appear twice) and deduce that the opposite faces of A and C are E
and F.
Finally B is opposite of D.
A
C
B
Adjacent faces Opposite face
B, C, D, F
E
B, D, A, E
F
D
Figure 2
A net diagram of the cube is shown in Figure 3.
F
B
A
D
E
C
Figure 3
Note : D appears twice in Figure 2..
A and E, B and D and C and F are known as opposite pairs.
When solving this problem, we apply deductive reasoning on two exclusive possibilities.
Any two labelled faces are either on adjacent sides or they are on opposite sides.
This problem is known as a deductive logic problem.
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Example 2
It is known that the six faces of each dice were labelled with 1, 2, 3, 4, 5
and 6 so that the sum of opposite numbers is equal to 7.
Figure 4 shows six jointed cubes such that the sum of the two numbers on
the jointed faces is equal to 6.
What is the number in “?" .
Solution
There are two rules:
The sum of opposite numbers is 7.
The sum of two numbers on the jointed faces is 6.
3
?
3
Figure 4
In Figure 4, we apply the two rules to get vertical pairs :
3
3+4=7
4
4+2=6
2
2+5=7
5
5+1=6
1
3
1+6=7
6
Figure 5
In the 3rd cube at the turning corner, the opposite of 3 is 4 and the remaining possible pair is either “2 and 5” or "5
and 2"as shown below.
First, we begin with 2 as the first number.
2
5
1
6
0
?
Figure 6
We can't apply the rule for the connecting faces between the 2rd and 3rd faces.
Next, we deduce with 5 the first number.
5
2
4
3
3
4
2
5
Figure 7
Therefore, "? = 5."
In this type of problems, we are either proceeding to make deductions with the rules or terminated by any of the
rules.
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2. Correct Labels
In the next problem, you will learn to use simple statements which are either Truth or False together with logical
deduction to solve problems.
Example 3
A mathematics teacher brought in three boxes: A, B and C. Each box is affixed with a label as shown in Figure 8.
Apple is in this
box
A
Apple is not in
this box
Apple is not in
this box A
B
C
Figure 8
An apple is inside one of the boxes and only one of the labels tells the truth.
Can you tell where the apple is?
Solution
Consider the conflicting statements:
The label on box A “Apple is in this box”
The label on box C “Apple is not in box A”
We can deduce that one of them is true and the other is false.
Labels
T
F
A
√√
√
B
×
√
C
√
√√
Figure 9
For the condition that only one of the labels tells the truth, we can further deduce that the label on box B “Apple is not
in this box”must be false and so the apple is inside the box B.
Remarks : In this problem, we make two deductions.
First we use two conflicting simple statements to make deduction.
Next, we make use of the given condition to make further deduction.
Example 4
There are two tennis balls in each of the three small boxes:
Figure 10
The labels posted outside the boxes are affixed incorrectly such that no label tells the actual contents of the box.
By looking at the color of a ball selected from a box, you can figure out the actual contents of the boxes. Which box
will you choose?
Solution
First, we construct a table to see different possibilities and drop
(×) some scenarios for the condition that no label tells the actual
contents of the box.
The box with mixed coloured label contains either
or
.By looking at the colore of one ball selected from the box
with mixed colored label, we can determine the actual contents
in the box and make further deduction on the actual contents in
the other two boxes as shown in the table.
What will happen if you open the box with “ ’label?
×
×
×
Figure 11
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