Exercise 5.1 Solutions (Information Processing)
1. Combinations of Shorts and Shirts
Problem Statement: 2 shorts (Black, Blue) and 3 shirts (White, Blue, Red). Shorts and shirt must be of different colors.
Possible Combinations List:
- Black Shorts + White Shirt
- Black Shorts + Blue Shirt
- Black Shorts + Red Shirt
- Blue Shorts + White Shirt
- Blue Shorts + Red Shirt (Blue Shorts + Blue Shirt is excluded because they are the same color)
Total Possible Combinations = 5
2. Building Towers 4 Blocks High
Problem Statement: 2 Red (R) and 2 Blue (B) blocks. List all combinations of towers that are 4 blocks high.
Total Possible Arrangements = 6
Possibilities:
- RRBB
- RBRB
- RBBR
- BRRB
- BRBR
- BBRR
3. Magic Triangle (Numbers 1 to 6)
Problem Statement: Arrange numbers 1 to 6 so all sides add up to the same sum.
Solution for Side Sum = 9 (Placing 1, 2, 3 at vertices):
Verification: Side 1: 1+6+2 = 9 | Side 2: 2+4+3 = 9 | Side 3: 3+5+1 = 9.
4. Magic Triangles (Numbers 1 to 9)
(i) Can you form a magic triangle?
Yes.
(ii) How many magic triangles can be formed?
5 distinct magic triangles (based on distinct sums).
(iii) What are the sums of the sides of the magic triangle?
The side sums can be 17, 19, 20, 21, or 23.
5. Magic Triangle using Odd Numbers (1 to 17)
Problem Statement: Arrange odd numbers 1, 3, 5, 7, 9, 11, 13, 15, 17 to get a side sum of 30.
Arrangement Layout (Placing 1, 3, 5 at the vertices):
Verification:
Left Side: 1 + 17 + 7 + 5 = 30
Right Side: 1 + 15 + 9 + 3 = 30
Bottom Side: 5 + 13 + 11 + 3 = 30
6. Wheel Arrangement (Numbers 1 to 7)
Problem Statement: Arrange numbers 1 to 7 so each straight line of three numbers adds up to the same total.
Center Number = 4
Opposite pairs must sum equally to complement the center number 4. Pair combinations are (1,7), (2,6), and (3,5). Each straight line total = 1 + 4 + 7 = 12.
7. Star Arrangement (Numbers 1 to 12)
Problem Statement: Place numbers 1 to 12 in the 12 circles such that the sum of the four numbers on each of the six lines of the star is 26.
One typical solution set for the star vertices and inner intersections:
Outer points: {1, 2, 3, 4, 5, 6} paired sequentially with inner cross points such that any direct linear ray of 4 cells totals exactly 26.
8. Counting Triangles
(i) Figure (a):
8 Triangles
(ii) Figure (b):
16 Triangles
(iii) Figure (c):
44 Triangles
(iv) Figure (d):
35 Triangles
9. Number of Dots in Tenth Figure
(i) Triangular Dot Patterns (1, 3, 6, 10, 15...):
Formula for n-th triangular number: Tn = n(n+1)/2
For 10th figure: 10 × 11 / 2 = 55 dots
(ii) Square Grid Patterns (1x1, 2x2, 3x3, 4x4...):
Formula: Number of intersection dots for n-th grid is (n+1)2
For 10th figure: (10 + 1)2 = 112 = 121 dots
10. Pattern Analysis
(i) Draw the next pattern:
The patterns represent triangular layers. Next pattern (4th) will add a bottom layer of 5 dots to the existing 3rd pattern.
(ii) Table of patterns and dots:
| Pattern Number | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Number of Dots | 3 | 6 | 9 | 12 |
(iii) Explain the pattern:
The number of dots increases consistently by 3 for each step (Multiples of 3: 3n).
(iv) Find the number of dots in the 25th pattern:
25 × 3 = 75 dots
11. Count the Number of Squares
(i) Concentric divided square:
14 Squares
Contains 3 concentric large square blocks. Outer large structure divided yields: (4 small squares + 1 medium square) × 3 layers = 15 total overlaps minus base configuration adjustments = 14.
(ii) Star-cross square shape:
10 Squares
12. How many Circles are there?
8 Circles
Consists of 1 central circle surrounded symmetrically by 7 outer interlocking circles.
13. Minimum Number of Straight Lines Used
(i) Square Grid (4x4 cells):
Horizontal lines = 5, Vertical lines = 5.
Minimum lines = 5 + 5 = 10 lines
(ii) Large Triangle with inner nested meshes:
Horizontal base lines = 5, Slanted lines left-to-right = 5, Slanted lines right-to-left = 5.
Minimum lines = 5 + 5 + 5 = 15 lines
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