Find every possible area for a square drawn by connecting dots of a 3 dot by 3 dot grid.
To find every possible area for a square drawn by connecting dots of a 3 dot by 3 dot grid, we can start by visualizing the grid and identifying all the possible squares.
A 3 dot by 3 dot grid looks like this:
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In this grid, we can have squares of different sizes, ranging from 1x1 (consisting of one dot) to 3x3 (the entire grid). Let's analyze the possible square sizes one by one:
1. 1x1 squares: Since a 1x1 square consists of only one dot, we can identify 9 such squares in the grid, as each dot can be the bottom left corner of a 1x1 square.
2. 2x2 squares: For a 2x2 square, we need to consider the combinations of four dots. Starting from the top left dot, we can build a 2x2 square by connecting the remaining three dots to its right and bottom. We can shift this square horizontally and vertically to find the other possible 2x2 squares. In total, there are four possible 2x2 squares.
3. 3x3 squares: The entire grid itself forms a 3x3 square, so there is only one such square.
Summing up the counts from each size of squares, we have:
9 (1x1 squares) + 4 (2x2 squares) + 1 (3x3 square) = 14 possible squares.
Therefore, there are 14 possible areas for a square drawn by connecting dots of a 3 dot by 3 dot grid.