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?

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

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

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

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