Math PicnicEvery sheet is a promise

Can it be done at all?

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The question

A shelf holds 19 glasses, and 1 of them is the right way up. You may turn over exactly 2 of them at a time, never more and never fewer.

What is the fewest turns that leaves every glass the right way up? Write NEVER if it cannot be done.

A floor, and a route that reaches it

now:1· want:19· each move:2
a route+a floor
8×2=16
18needed
9×2=18
9
  1. 19 glasses, and the number standing the right way up is 1. Every move has to touch exactly 2 of them — never more and never fewer — and the target is to get the number standing the right way up to 19: every one of the 19.
  2. "Fewest" needs two things: a way that works, and a reason nothing shorter can. Start with the second.
  3. One turn touches 2 glasses, so at the very best it moves the number standing the right way up by 2. There are 19 − 1 = 18 to make up, and 8 turns could shift at most 16 — not enough.
  4. So at least 9. Can 9 be done? 9 × 2 = 18, which is exactly what is needed.
  5. So 9 turns: one fewer is impossible, and 9 can be done.

How it works.

  1. 01

    19 glasses, and the number standing the right way up is 1. Every move has to touch exactly 2 of them — never more and never fewer — and the target is to get the number standing the right way up to 19: every one of the 19.

  2. 02

    "Fewest" needs two things: a way that works, and a reason nothing shorter can. Start with the second.

  3. 03

    One turn touches 2 glasses, so at the very best it moves the number standing the right way up by 2. There are 19 − 1 = 18 to make up, and 8 turns could shift at most 16 — not enough.

  4. 04

    So at least 9. Can 9 be done? 9 × 2 = 18, which is exactly what is needed.

  5. 05

    So 9 turns: one fewer is impossible, and 9 can be done.

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