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Answer :
The number of black squares in a 25x25 grid painted in a spiral pattern is 625.
Grid Size
The grid is 25x25, meaning it has 625 squares in total.
2. Center and Spiral Pattern
We start at the center, which is at position (13,13) in a 1-based index.
The painting starts with black at the center and spirals outwards.
The spiral pattern results in concentric squares of increasing size. Each concentric square can be represented as [tex](2n+1) \times (2n+1)[/tex]
n=0 gives the center square, which is [tex]1 \times 1[/tex].
n=1 gives the next layer, which is a [tex]3 \times 3[/tex] square surrounding the center.
n=2 gives a [tex]5 \times 5[/tex] square, and so on.
Summing the Series of Odd Numbers
The total number of black squares follows the pattern of summing the areas of these concentric squares:
For n=0
The area is 1²= 1
For n=1
The area is 3² = 9
For n=2
The area is 5² = 25
And so on, up to n=12 because [tex]2 \times 12 + 1 = 25[/tex], the side length of the outermost square.
Summing the Areas
Sum the squares of the first 13 odd numbers (since n ranges from 0 to 12)
[tex]1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2 + 13^2 + 15^2 + 17^2 + 19^2 + 21^2 + 23^2 + 25^2[/tex]
The sum of the first k odd numbers is given by k²
k = 13 (since we are counting up to the 25th odd number).
Thus, the sum of the first 13 odd numbers is 13² = 169
This represents the total number of black squares in each concentric square up to the edge of the 25x25 grid.
Given this series and the pattern of summing the odd numbers, we conclude:
625 squares .
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Rewritten by : Jeany
