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Number Play - Supercells

Grade 6CBSE

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

🔑Concepts

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A Magic Square is an n×nn \times n grid filled with distinct numbers such that the sum of the numbers in each row, each column, and both main diagonals is the same.

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The constant sum is known as the Magic Constant. For a 3×33 \times 3 magic square, if the middle number is mm, the Magic Constant is 3m3m.

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In a 3×33 \times 3 magic square using consecutive numbers from 11 to 99, the Magic Constant is 1515, and the middle number is 55.

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Supercells usually refer to 2×22 \times 2 or 3×33 \times 3 sub-grids within a larger number chart (like a 10×1010 \times 10 number grid). The sum of numbers in these supercells often follows predictable mathematical patterns.

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In a 10×1010 \times 10 number grid, the sum of a 3×33 \times 3 supercell is always 9×the central number9 \times \text{the central number} of that supercell.

📐Formulae

S=n(n2+1)2S = \frac{n(n^2 + 1)}{2}

S=3×Middle Number (for 3×3 magic squares)S = 3 \times \text{Middle Number} \text{ (for } 3 \times 3 \text{ magic squares)}

Sum of 2×2 supercell=4×Average of the four numbers\text{Sum of } 2 \times 2 \text{ supercell} = 4 \times \text{Average of the four numbers}

Sum of 3×3 supercell=9×Central Number\text{Sum of } 3 \times 3 \text{ supercell} = 9 \times \text{Central Number}

💡Examples

Problem 1:

Determine the Magic Constant for a 3×33 \times 3 magic square using the numbers 7,8,9,10,11,12,13,14,157, 8, 9, 10, 11, 12, 13, 14, 15.

Solution:

The given sequence of numbers is 7,8,9,10,11,12,13,14,157, 8, 9, 10, 11, 12, 13, 14, 15. The middle number in this sequence is 1111. Magic Constant S=3×11=33S = 3 \times 11 = 33.

Explanation:

In a 3×33 \times 3 magic square consisting of numbers in an arithmetic progression, the magic constant is always three times the middle term.

Problem 2:

In a 10×1010 \times 10 number grid (1 to 100), find the sum of a 3×33 \times 3 supercell where the central number is 4545.

Solution:

Central number =45= 45. Sum of 3×33 \times 3 supercell =9×45=405= 9 \times 45 = 405.

Explanation:

For any 3×33 \times 3 grid within a standard number chart, the total sum is 9 times the value of the cell located at the center.

Problem 3:

Calculate the sum of the following 2×22 \times 2 supercell from a number grid: 24253435\begin{array}{r} 24 & 25 \\ 34 & 35 \end{array}

Solution:

242534+35118\begin{array}{r} 24 \\ 25 \\ 34 \\ + 35 \\ \hline 118 \end{array} The sum is 118118.

Explanation:

We add the four numbers in the 2×22 \times 2 block: 24+25+34+35=11824 + 25 + 34 + 35 = 118.