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Patterns - Magic Squares and Triangles

Grade 5CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Magic Square is a square grid where the sum of numbers in each row, each column, and both main diagonals is the same constant value. Visually, a 3×33 \times 3 magic square looks like a grid of 9 cells (3 rows and 3 columns) where every horizontal, vertical, and corner-to-corner line of three numbers adds up to the same total.

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The Magic Sum (or Magic Constant) is the specific total that every row, column, and diagonal must equal. For a 3×33 \times 3 magic square using consecutive numbers, the magic sum is always exactly 3 times the number placed in the center cell.

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In a 3×33 \times 3 magic square, the layout follows a specific balance. If the grid is filled with consecutive numbers, the middle number of the sequence must occupy the center cell to maintain symmetry and ensure all sums are equal.

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A Magic Triangle is a triangular arrangement of numbers where the sum of numbers on each of the three sides is equal. Visually, this is represented by circles placed at the three vertices (corners) and circles placed along the edges between those corners.

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The sum of a Magic Triangle side depends heavily on the 'Corner Numbers.' Because each corner number is part of two different sides, these numbers are added twice when calculating the total of all sides. To achieve a larger side sum, larger numbers should be placed at the corners; to achieve a smaller side sum, smaller numbers should be placed at the corners.

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Pattern Discovery in Triangles: When using numbers 1 to 6 in a triangle (3 numbers per side), the total sum of all three sides is calculated as Sum of numbers (1+2+3+4+5+6)+Sum of corner numbersSum \text{ } of \text{ } numbers \text{ } (1+2+3+4+5+6) + Sum \text{ } of \text{ } corner \text{ } numbers. This helps in identifying which numbers must go into the corners to reach a target sum.

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Solving for missing numbers involves the Subtraction Method. If you know the Magic Sum and two numbers in a row or on a side, you can find the third number by subtracting the sum of the two known numbers from the Magic Sum.

📐Formulae

Magic Sum of 3×3 Square=3×Middle Number\text{Magic Sum of } 3 \times 3 \text{ Square} = 3 \times \text{Middle Number}

Total Sum of all sides (Triangle)=(Sum of all numbers used)+(Sum of corner numbers)\text{Total Sum of all sides (Triangle)} = (\text{Sum of all numbers used}) + (\text{Sum of corner numbers})

Magic Constant (n × n square)=n(n2+1)2\text{Magic Constant (n } \times \text{ n square)} = \frac{n(n^2 + 1)}{2}

Missing Number=Magic Sum−(Sum of known numbers in that line)\text{Missing Number} = \text{Magic Sum} - (\text{Sum of known numbers in that line})

💡Examples

Problem 1:

Complete a 3×33 \times 3 magic square using numbers from 46 to 54. The Magic Sum is 150.

Solution:

  1. Find the middle number of the sequence 46, 47, 48, 49, 50, 51, 52, 53, 54. The middle number is 5050.
  2. Place 5050 in the center cell.
  3. Check the rule: Magic Sum=3×Middle Number⇒150=3×50\text{Magic Sum} = 3 \times \text{Middle Number} \Rightarrow 150 = 3 \times 50. This is correct.
  4. Place the remaining numbers such that each row, column, and diagonal adds to 150.
  5. Top row: 53+46+51=15053 + 46 + 51 = 150
  6. Middle row: 48+50+52=15048 + 50 + 52 = 150
  7. Bottom row: 49+54+47=15049 + 54 + 47 = 150
  8. Check diagonal: 53+50+47=15053 + 50 + 47 = 150.

Explanation:

By placing the median of the sequence in the center and balancing the largest numbers with the smallest numbers on opposite sides, we satisfy the magic sum requirement for all directions.

Problem 2:

Arrange the numbers 1, 2, 3, 4, 5, and 6 in a Magic Triangle so that the sum of each side is 9.

Solution:

  1. Sum of all numbers provided: 1+2+3+4+5+6=211 + 2 + 3 + 4 + 5 + 6 = 21.
  2. Target sum for 3 sides: 9×3=279 \times 3 = 27.
  3. Find the sum of corner numbers: Total target−Sum of numbers=27−21=6\text{Total target} - \text{Sum of numbers} = 27 - 21 = 6.
  4. Identify three numbers from the set {1, 2, 3, 4, 5, 6} that add up to 6. These are 1,2, and 31, 2, \text{ and } 3.
  5. Place 1, 2, and 3 at the corners of the triangle.
  6. Find the middle numbers for each side:
    • Side between corners 1 and 2: 9−(1+2)=69 - (1 + 2) = 6.
    • Side between corners 2 and 3: 9−(2+3)=49 - (2 + 3) = 4.
    • Side between corners 3 and 1: 9−(3+1)=59 - (3 + 1) = 5.
  7. The sides are (1,6,2)(1, 6, 2), (2,4,3)(2, 4, 3), and (3,5,1)(3, 5, 1). All sum to 9.

Explanation:

The 'extra' sum needed to reach the side totals comes from the corner numbers being counted twice. By calculating that the corners must sum to 6, we correctly identify 1, 2, and 3 as the vertex numbers.