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### When is a Circle a Square? (Weekly Brain Potion)

When is a Circle a Square? (Weekly Brain Potion)

Gloria has built her first robot, named Robbie. Since it is her first robot, there are some quirks in Robbie's behavior.

On of these quirks is Robbie's movement. He can only move vertically and horizontally (different from Gloria who can move diagonally as well). Because of his different movement, Robbie and Gloria have different notions of distance. For example, when tasked from moving from the points A to B shown below, Robbie says he will need to move $7$ units while Gloria says she only needs to move $5$ units.

Gloria begins to explore their two ideas of distance. Remembering her geometry class, she draws a circle with center A and radius $5$ which contains point $B$ (the circle is shown below). That is, she draws the collection of points that are distance $5$ from A.

Can you help Gloria figure out what Robbie's 'circle' will look like? That is, what is the collection of points that are distance $7$ from A according to Robbie?

Fascinated by these two ideas of distance, Gloria thinks of some properties we take for granted with distance. Things such as

• The distance is never negative, and zero only if you don't move.
• The distance from P to Q is the same as the distance from Q to P.
• The distance from P to Q is never more than the distance from P to R plus the distance from R to Q.

Are all of these still true for Robbie? What other differences can you find between Gloria's distance and Robbie's distance?