Question

Veronica made the pattern shown out of 7 rectangles with four equal sides. The side length of each rectangle is written inside the rectangle. How many rectangles can be formed using the lines in this figure? (Hint: The answer is not 7.)

Answer
Attempt 2 out of 3

Answer 1

I don’t see the picture — could you upload the diagram or describe exactly how the 7 squares are arranged (rows/columns or a sketch)? I can’t give the final count without that.

Meanwhile, here’s a short, reliable method you can use (or I can apply once I see the figure):

Method for counting all rectangles whose sides lie on the given grid lines
- Treat the pattern as made from unit squares on a square grid (since each “rectangle with four equal sides” is a square).
- List the distinct vertical grid lines and horizontal grid lines formed by the pattern.
- For every choice of two vertical lines (left and right), look at the rows between them and mark which rows have squares filling every column between those two verticals. Each contiguous run of r such rows gives r(r+1)/2 different vertical choices (i.e., different heights) for rectangles with those left/right boundaries.
- Sum that count over all choices of left and right vertical lines. That counts every axis-aligned rectangle (including squares) exactly once.

Example (quick): a full 3×3 block has 4 vertical and 4 horizontal grid lines, so number of rectangles = C(4,2)*C(4,2) = 6*6 = 36. For irregular shapes you must do the “pairs of vertical lines + contiguous filled rows” tally above.

If you upload the image or describe the arrangement (e.g., coordinates or a small ASCII sketch), I’ll count them precisely.