Math PicnicEvery sheet is a promise

Can it be done at all?, step by step

Twelve cups stand upside down.

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The question

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" always wants two things and most answers only give one: a way that works, and a reason nothing shorter can. Do the second one first, because it is easier here and it tells you what to aim for.
  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 9 is the floor, and a floor is only worth having if something reaches it. 9 × 2 = 18, which is exactly what is needed, so every move can count for its full value.
  5. 9 turns, then — one fewer is impossible and 9 is achievable, which is both halves of the answer rather than just the half that gets written.

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" always wants two things and most answers only give one: a way that works, and a reason nothing shorter can. Do the second one first, because it is easier here and it tells you what to aim for.

  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 9 is the floor, and a floor is only worth having if something reaches it. 9 × 2 = 18, which is exactly what is needed, so every move can count for its full value.

  5. 05

    9 turns, then — one fewer is impossible and 9 is achievable, which is both halves of the answer rather than just the half that gets written.

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