The three utilities puzzle with math/science YouTubers

The video features a group of YouTubers attempting a puzzle involving connecting utilities to houses without crossing lines. The challenge is proven impossible on a plane using mathematical reasoning, but solvable on a mug due to its topology. The video explains the mathematical concepts involved, including Euler's formula, and concludes with a sponsor message.

What is the relationship between the number of vertices, edges, and regions in a planar graph?

How does the speaker suggest using the mug to avoid crossing lines?

What is the outcome of the speaker's attempt to solve the puzzle?

What does the speaker encourage viewers to do when faced with challenging puzzles?