2021 AMC 12A Problems/Problem 4

The following problem is from both the 2021 AMC 10A #7 and 2021 AMC 12A #4, so both problems redirect to this page.

Problem

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?

$\textbf{(A) }$ Purple snakes can add.

$\textbf{(B) }$ Purple snakes are happy.

$\textbf{(C) }$ Snakes that can add are purple.

$\textbf{(D) }$ Happy snakes are not purple.

$\textbf{(E) }$ Happy snakes can't subtract.

Solution 1

We know that purple snakes cannot subtract, thus they cannot add either. Since happy snakes must be able to add, the purple snakes cannot be happy. Therefore, we know that the happy snakes are not purple and the answer is $\boxed{\textbf{(D)}}$ .

--abhinavg0627

Solution 2 (Arrows)

We are given that

$\text{(1) Happy}\Rightarrow\text{can add}$

$\text{(2) Purple}\Rightarrow\text{cannot subtract}$

$\text{(3) Cannot subtract}\Rightarrow\text{cannot add}$

Combining $\text{(2)}$ and $\text{(3)}$ into $\text{(*)}$ below, we have

$\text{(1) Happy}\Rightarrow\text{can add}$

$\text{(*) Purple}\Rightarrow\text{cannot subtract}\Rightarrow\text{cannot add}$

Clearly, the answer is $\boxed{\textbf{(D)}}.$

~MRENTHUSIASM

Video Solution by Aaron He (Sets)

https://www.youtube.com/watch?v=xTGDKBthWsw&t=284

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=MUHja8TpKGw&t=259s (Note that there's a slight error in the video I corrected in the description)

Video Solution by Hawk Math

https://www.youtube.com/watch?v=P5al76DxyHY

Video Solution (Using logic to eliminate choices)

https://youtu.be/Mofw3VXHPyg

~ pi_is_3.14

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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