Daily Math Puzzle: 2026-07-17
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2026-07-17
A shopper is trying to fit as many cylindrical cans of soup as possible into a single layer at the bottom of a rectangular shopping basket. Each can has a diameter of 7 cm. The internal base dimensions of the basket are 30 cm long by 20 cm wide. How many cans can they fit?
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Solution
11 cans — This puzzle requires spatial reasoning about packing circles (cans) into a rectangular space. The trick lies in recognizing the most efficient packing method.
**Less Efficient Approach: Grid Packing**
If the cans are placed in a simple, non-staggered rectangular grid:
- Along the 30 cm length: 30 cm / 7 cm per can = 4.28... This means 4 cans can fit (4 * 7 cm = 28 cm, with 2 cm space remaining).
- Along the 20 cm width: 20 cm / 7 cm per can = 2.85... This means 2 cans can fit (2 * 7 cm = 14 cm, with 6 cm space remaining).
Total cans: 4 cans (length) * 2 cans (width) = 8 cans.
**More Efficient Approach: Hexagonal Packing (Staggered Rows)**
For packing circular objects, a hexagonal (staggered) arrangement is more efficient than a simple grid because it minimizes the wasted space between circles.
First, determine how many rows can fit along the 20 cm width:
- The first row of cans takes up 7 cm of width.
- Each subsequent staggered row adds a vertical distance of (diameter * √3 / 2) from the previous row. (7 cm * √3 / 2) ≈ 6.06 cm.
- **Row 1:** Takes 7 cm. (Fits within 20 cm).
- **Row 2 (staggered):** Adds 6.06 cm. Total width for 2 rows: 7 cm + 6.06 cm = 13.06 cm. (Fits within 20 cm).
- **Row 3 (staggered):** Adds another 6.06 cm. Total width for 3 rows: 13.06 cm + 6.06 cm = 19.12 cm. (Fits within 20 cm).
- **Row 4 (staggered):** Would require 19.12 cm + 6.06 cm = 25.18 cm, which exceeds the 20 cm basket width. So, only 3 rows can fit.
Next, determine the number of cans per row along the 30 cm length:
- For the 'full' rows (Row 1 and Row 3), the number of cans is floor(30 cm / 7 cm) = 4 cans each.
- For the 'staggered' row (Row 2), the cans are offset. If a full row fits 4 cans with 2cm space (28cm total), a staggered row (where the centers align with the gaps of the previous row) will effectively fit one less can. So, Row 2 can fit 3 cans.
Total cans: 4 (Row 1) + 3 (Row 2) + 4 (Row 3) = 11 cans.
Therefore, by arranging the cans hexagonally, a maximum of 11 cans can fit in a single layer.
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