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The Carpenter and the Nails

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A 34" x 34" square metal plate needs to be fixed by a carpenter on to a wooden board. The carpenter uses nails all along the edges of the square such that there are 35 nails on each side of the square. Each nail is at the same distance from the neighboring nails. How many nails does the carpenter use?
Answer: 136

Solution:

The four corners of the square have one nail each.

Each side of the square now requires 33 nails (excluding the four corners).

Thus, total number of nails used = 4 + (4 × 33) = 136.

Food for thought:

What if the metal plate is in the shape of a regular polygon (e.g., pentagon, hexagon, heptagon, octagon, etc.)? Try generalizing the above result to the case of a regular polygon.

Is the information that the square is 34" x 34" required? It is important in real-world problem-solving to recognize information that is "extra" or "missing".


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