Problem 23 worked answer

Grogu’s Force Sort

Designed for Grade 2 · Grade 3 · Uses Middle Grade 1 math · About 4–11 minutes

Earn one jewelMarble TabletLevel 3

At a glance

  • Part 1: From left to right, the only order is blue cookie, silver control knob, soup cup, frog.
  • Part 2: The Mandalorian is not right. Without Clue 3, blue cookie, frog, silver control knob, soup cup is a different order that obeys Clues 1, 2, and 4.

Key idea: start with the tightest pair

Clue 2 locks the silver control knob and soup cup together, in that order. Clue 4 says the knob cannot be at an end. Therefore the knob–cup pair has only two possible places: positions 2–3 or positions 3–4.

Testing those two cases is enough to find the arrangement and prove it is unique.

1. Find the only order

First try the silver control knob in position 3. Clue 2 then puts the soup cup immediately to its right in position 4.

Clue 3 says the frog is not next to the knob. It cannot be in position 2, and positions 3 and 4 are already used, so the frog would have to be in position 1. But Clue 1 requires the blue cookie to be somewhere left of the frog. Nothing is left of position 1, so this case is impossible.

The silver control knob must therefore be in position 2. Clue 2 puts the soup cup in position 3.

The frog cannot be next to the knob. It cannot use position 1 or position 3, so it must use position 4. The only position left for the blue cookie is position 1.

The order is:

  1. Blue cookie
  2. Silver control knob
  3. Soup cup
  4. Frog

Check every clue

  • The blue cookie in position 1 is left of the frog in position 4.
  • The soup cup in position 3 is immediately to the right of the knob in position 2.
  • The frog in position 4 is not next to the knob in position 2.
  • The knob in position 2 is not at either end.

The arrangement obeys all four clues. It is unique because Clue 4 leaves only positions 2 and 3 for the knob, and the position-3 case was impossible.

2. Test whether Clue 3 is needed

Ignore only Clue 3 and try this different order:

  1. Blue cookie
  2. Frog
  3. Silver control knob
  4. Soup cup

This order obeys the remaining clues:

  • Clue 1 works because the cookie is left of the frog.
  • Clue 2 works because the soup cup is immediately to the right of the knob.
  • Clue 4 works because the knob is in position 3, not at an end.

It breaks only Clue 3 because the frog is next to the knob. Once Clue 3 is removed, both this order and the Part 1 order are possible. Therefore Clue 3 is needed to make the answer certain.

Technical fit and rating

MJ 1.3 · C3 · W2 Stretch challenge · Moderate workload

Geometry

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