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2000 AMC 10 Problem 21

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2000 AMC 10 Problem 21AMC 10 Step by Step
Name:                             Date: September 13, 2026
Logic Puzzles · difficulty 3 of 5 · about 3 min · position P21-25

If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?

I. All alligators are creepy crawlers.\textrm{I. All alligators are creepy crawlers.} II. Some ferocious creatures are creepy crawlers.\textrm{II. Some ferocious creatures are creepy crawlers.} III. Some alligators are not creepy crawlers.\textrm{III. Some alligators are not creepy crawlers.}

  1. A)

    I only\text{I only}

  2. B)

    II only\text{II only}

  3. C)

    III only\text{III only}

  4. D)

    II and III only\text{II and III only}

  5. E)

    None must be true\text{None must be true}

Answer

B

Key insight

Only statement II follows: the creepy crawlers that are alligators are automatically ferocious; I and III each fail in a small made-up world.

Solution

Abbreviate: every AA (alligator) is an FF (ferocious); at least one CC (creepy crawler) is an AA.

Statement II: take a creature that is both CC and AA. Being an AA, it is an FF. So there is a creature that is both FF and CC, and II must be true.

Statement I: build a world with two alligators, only one of which is a creepy crawler. Both hypotheses hold, yet I is false, so I is not forced.

Statement III: build a world with exactly one alligator, and it is a creepy crawler. Both hypotheses hold, yet III is false, so III is not forced.

The answer is (B) II only\boxed{\textbf{(B)}\ \text{II only}}.

Why this works

"Must be true" means true in every world consistent with the hypotheses. To confirm a statement, derive it; to reject one, exhibit a single tiny world where the hypotheses hold and the statement fails. In these problems "some" means "at least one" and never rules out "all."

The trap

Importing real-world facts about alligators, or reading 'some' as 'some but not all'.

Common mistakes

  • Importing real-world facts about alligators, or reading 'some' as 'some but not all'.
  • Believing statements I and III cannot both be false, when a world with one creepy-crawler alligator falsifies III and a second non-crawler alligator falsifies I.

Techniques

Test small/specific values or special cases to find or verify the answer