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2021 AMC 10A Problem 7

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2021 AMC 10A Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Logic Puzzles · difficulty 2 of 5 · about 2 min · position P1-10

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that

all of his happy snakes can add,

none of his purple snakes can subtract, and

all of his snakes that can't subtract also can't add.

Which of these conclusions can be drawn about Tom's snakes?

A)
$Purple snakes can add.$
B)
$Purple snakes are happy.$
C)
$Snakes that can add are purple.$
D)
$Happy snakes are not purple.$
E)
$Happy snakes can't subtract.$

Answer

Key insight

Chain the implications: purple -> cannot subtract -> cannot add, while happy -> can add; so a happy snake cannot be purple.

Solution

Write the three facts as implications:
1. happy $\Rightarrow$ can add;
2. purple $\Rightarrow$ cannot subtract;
3. cannot subtract $\Rightarrow$ cannot add.

Chaining 2 and 3: purple $\Rightarrow$ cannot add. Combining with 1, a happy snake can add, so it is not one of the snakes that cannot add, so it is not purple. That is exactly statement (D).

The other choices fail: (A) contradicts the chain just derived; (B) contradicts (D) unless there are no purple snakes, but there are four; (C) reverses fact 1's direction and is not implied; (E) is not implied, since nothing says a snake that can add cannot subtract.

The answer is $\boxed{\textbf{(D)}\ \text{Happy snakes are not purple.}}$

Why this works

Every "all X are Y" is an implication $X \Rightarrow Y$ , and implications chain: $A \Rightarrow B$ and $B \Rightarrow C$ give $A \Rightarrow C$ . The contrapositive ( $\lnot C \Rightarrow \lnot A$ ) is equally valid, and that is what produces "happy $\Rightarrow$ not purple." The numbers $13$ , $4$ , $5$ are pure decoration.

The trap

Reversing an implication, e.g. reading 'all happy snakes can add' as 'all snakes that can add are happy' (choice C).

Common mistakes

Reversing an implication, e.g. reading 'all happy snakes can add' as 'all snakes that can add are happy' (choice C).
Choosing (E) by assuming every snake either adds or subtracts but not both; the statements never say that.

Techniques

Set up the equation/formula and compute; no special trick needed