## Game Board

**Taxicab geometry** is a form of geometry in which you can only move along the lines of a grid. Taxicab geometry gets its name from taxis since they can only drive along streets rather than moving as the bird flies.

## Instructions

- Player 1 chooses a point to hide on the coordinate plane. Player 2 makes their first guess to find the hidden point by telling an ordered pair.

- The first player then uses their opponent's guess point to write the taxicab ( ← ↑ → ↓) distance between their hiding point and the guess point.

- The second player can mark the points which have the same distance to their guess point on the coordinate plane using the point tool. Then tells another ordered pair as their second guess.

4. If the player 1 cannot be found in 5 guesses, they are free to change their hiding place!

## Discussion Questions

After the first game, discuss the properties of the Taxicab Geometry with students using the following questions;

- In Euclidian Geometry, we define the circles as the set of points equidistant to a certain point (center). So what would be the circles of Taxicab Geometry?

- What would be the value of $π$ in the Taxicab World?

- What is the taxicab distance function between the points $(x_1,y_1)$ and $(x_2,y_2)$?

## Solution

The taxicab distance function is the sum of the absolute differences of their Cartesian coordinates.

$d = |x_1 - x_2| + |y_1 - y_2|$

The circles of the taxicab geometry are squares; therefore the value of π would be 4. Click here to read more about π in squares.