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Number Play - Some Explorations in Grids

Grade 7CBSE

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

🔑Concepts

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A grid is a square or rectangular arrangement of numbers. An n×nn \times n grid contains n2n^2 cells.

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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 constant value, known as the Magic Constant.

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In any 3×33 \times 3 grid extracted from a calendar or a consecutive number grid, the sum of all nine numbers is always 99 times the middle number.

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In a 2×22 \times 2 square within a number grid, the sum of the numbers on one diagonal is equal to the sum of the numbers on the other diagonal: a+d=b+ca + d = b + c, where (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} represents the grid.

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For a 3×33 \times 3 magic square using numbers 11 to 99, the middle number is always 55 and the magic constant is 1515.

📐Formulae

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

Sum of numbers in a 3×3 grid=9×Middle Number\text{Sum of numbers in a } 3 \times 3 \text{ grid} = 9 \times \text{Middle Number}

Total sum of first N numbers=N(N+1)2\text{Total sum of first } N \text{ numbers} = \frac{N(N + 1)}{2}

💡Examples

Problem 1:

Calculate the Magic Constant for a 4×44 \times 4 magic square using numbers from 11 to 1616.

Solution:

Using the formula for the Magic Constant where n=4n = 4: S=4(42+1)2S = \frac{4(4^2 + 1)}{2} S=4(16+1)2S = \frac{4(16 + 1)}{2} S=2×17=34S = 2 \times 17 = 34

Explanation:

The Magic Constant SS represents the sum that every row, column, and diagonal must add up to in an n×nn \times n magic square.

Problem 2:

In a calendar, a 3×33 \times 3 block of dates is chosen. The middle date is 1414. Find the total sum of all the dates in this block.

Solution:

For a 3×33 \times 3 grid in a calendar, the sum is given by: Sum=9×Middle Number\text{Sum} = 9 \times \text{Middle Number} Sum=9×14=126\text{Sum} = 9 \times 14 = 126

Explanation:

Because the numbers in a calendar grid follow an arithmetic pattern, the average of the nine numbers is exactly the middle number.

Problem 3:

Consider a 2×22 \times 2 grid from a number chart: (12132223)\begin{pmatrix} 12 & 13 \\ 22 & 23 \end{pmatrix} Verify the diagonal sum property.

Solution:

Sum of first diagonal: 12+2335\begin{array}{r} 12 \\ + 23 \\ \hline 35 \end{array} Sum of second diagonal: 13+2235\begin{array}{r} 13 \\ + 22 \\ \hline 35 \end{array} Since 12+23=13+22=3512 + 23 = 13 + 22 = 35, the property holds.

Explanation:

In any 2×22 \times 2 square in a grid where numbers increase by a constant 11 horizontally and a constant kk vertically, the sums of the diagonal elements are always equal.