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How many matchsticks make a row of 1000 octagons?

The rule: count what the first shape uses, then what each new shape adds

When identical polygons are joined in a row and neighboring polygons share one full side, the first polygon uses all its sides. Every later polygon reuses one existing side, so it contributes one fewer new stick. For $n$ octagons, the first uses $8$ sticks and the remaining $n - 1$ octagons add $7$ each:

S_{n} = 8 + 7 (n - 1) = 7 n + 1

Problem

A row of $1000$ regular octagons is made from equal matchsticks. Each octagon after the first shares one complete side with the previous octagon. How many matchsticks are used altogether?

Three octagons in a row. Each neighboring pair shares one side, so a new octagon adds 7 sticks rather than 8.

Answer

7001

matchsticks.

Checked two ways: incremental construction and correction for shared edges.

Step-by-step solution

1. Start with one octagon

One octagon has $8$ sides, so it needs $8$ matchsticks.

2. Add the remaining octagons

A new octagon shares one side already present in the row. It therefore needs only $7$ new sticks. There are $1000 - 1 = 999$ octagons after the first.

S = 8 + 999 \cdot 7

S = 8 + 6993 = 7001

A second way to verify it

If the octagons were separate, they would use $8 \cdot 1000 = 8000$ sticks. The row has $999$ joins. At each join, two separately counted sides become one physical stick, so subtract $999$ :

8000 - 999 = 7001

Both methods agree.

General formula

For $n$ octagons in such a row:

S_{n} = 8 + 7 (n - 1) = 7 n + 1

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