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Exploring Non-Euclidean Shortest Paths on a Grid: A Taxi-Cab Geometry Problem

This problem explores the concept of shortest distances in a non-Euclidean geometry context using a grid system with 1-inch-square edges. It examines the shortest path between points A and B, requiring the path to stay on grid lines. You will verify your findings with two drawings and demonstrate there are no shorter distances. Additionally, the task involves identifying the number of shortest paths available. By analyzing simpler problems, deeper patterns emerge regarding the shortest path lengths and their variations in 1x, 2x, and 3x grid systems.

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Exploring Non-Euclidean Shortest Paths on a Grid: A Taxi-Cab Geometry Problem

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  1. Taxi Cab Geometry Non-Euclidian Counting Problem

  2. What is the shortest distance? • What is the shortest distance from A to B? Let each edge of a square be 1 inch.

  3. What is the shortest distance? • What is the shortest distance from A to B if you have to stay on the grid lines? • Verify by having 2 drawings. Explain how you know there isn’t a shorter distance.

  4. How many shortest paths are there?

  5. Problem Solving Skills • Study a simpler problem and look for patterns.

  6. 1x systems • For the 1x blocks, find the length of a shortest path and how many shortest paths there are. Note any patterns.

  7. 2x systems • For the 2x blocks, find the length of a shortest path and how many shortest paths there are. Note any patterns.

  8. 3x systems • For the 3x blocks, find the length of a shortest path and how many shortest paths there are. Note any patterns.

  9. How many shortest paths are there?

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