Walls of Rock

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Beaumaris castle in Wales was the largest of the Welsh castles built by King Edward I (r. 1272-1307) Construction began in 1295, but was never completed according to the original plans.

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Medieval castles like Beaumaris were often built with a roughly rectangular design.

Generally, a larger area was worth protecting with thicker walls, so the larger castles tend to have thicker walls.

Below, a castle’s courtyard of area 24 meters squared is protected by rock walls of thickness 2 meters. The walls cover 56 meters squared.

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Is it possible for the area of the rock walls to equal the area of the castle’s courtyard if the thickness of the castle walls is constant?

Extensions:

  • Is there more than one such castle?
  • Is there a solution for any thickness of wall?
  • Is there a solution for any ratio of length to width?
  • If you know the perimeter of the castle and the thickness of the walls, can you calculate the area of the courtyard?

3D Extension:

  • Design and solve similar problems for a space station.

The Math in This Problem:

The Walls of Rock puzzle proposes the problem of constructing a large castle, including thick walls as well as a big courtyard area. Using the knowledge they have of areas and other mathematical operations, students are challenged to find a way to reconstruct the castle to have the two areas equal, while keeping the thickness of the walls constant.

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