Math PicnicEvery sheet is a promise

The spider and the fly, step by step

A spider in one corner of a room, a fly in the far opposite corner.

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The question
An ant is at one corner of a 5 × 10 × 20 box and wants to reach the corner diagonally opposite. It can only crawl along the outside faces — it cannot cut through the box.What is the shortest distance across the surface?52010Why

What is the shortest distance across the surface?

Unfold the box, then go straight

box:5 × 10 × 20
5+10=15
15×15=225
20×20=400
25×25=625
shortest:25
  1. A bug crawls the OUTSIDE of an 5 × 10 × 20 box, corner to the far opposite corner, and may not cut through. Nothing is measured yet.
  2. A bent path over two faces becomes a straight line once those faces are unfolded flat. Pairing the two shorter edges lays them along one side: 5 + 10 = 15, with 20 the other side.
  3. Pythagoras on that flat rectangle: 15 × 15 = 225 and 20 × 20 = 400, together 625; and 25 × 25 = 625, so this path is 25.
  4. It is the shortest: the other two unfoldings come out longer (squared lengths 725 and 925), and cutting straight through the box is shorter still but leaves the surface, so it does not count. Shortest surface path: 25.

How it works.

  1. 01

    A bug crawls the OUTSIDE of an 5 × 10 × 20 box, corner to the far opposite corner, and may not cut through. Nothing is measured yet.

  2. 02

    A bent path over two faces becomes a straight line once those faces are unfolded flat. Pairing the two shorter edges lays them along one side: 5 + 10 = 15, with 20 the other side.

  3. 03

    Pythagoras on that flat rectangle: 15 × 15 = 225 and 20 × 20 = 400, together 625; and 25 × 25 = 625, so this path is 25.

  4. 04

    It is the shortest: the other two unfoldings come out longer (squared lengths 725 and 925), and cutting straight through the box is shorter still but leaves the surface, so it does not count. Shortest surface path: 25.

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