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 cant subtract also cant add
Which of these conclusions can be drawn about Toms snakes?
Answer Choices:
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
Solution:
To see that choices (A), (B), (C), and (E) do not follow from the given informtion, consider the following two snakes that may be part of Tom's collection. One snake is happy but not purple and can both add and subtract. The second is purple but not happy and can neither add nor subtract. Then each of the three bulleted statements is true, but each of these choices is false.
To show that answer choice (D) is correct, first observe that the third bulleted statement is equivalent to "Snakes that can add also can subtract." The second bulleted statement is equivalent to "Snakes that can subtract are not purple." The three bulleted statements combined then lead to the conclusion "Happy snakes are not purple."
OR
Let , and denote the statements that a snake is happy, is purple, can add, and can subtract, respectively. Let denote "implies" and denote "not". Recall that an implication is logically equivalent to its contrapositive
Then the three bulleted statements can be exactly summarized as
Choice (D), which is
follows from the transitivity of the "implies" relation. However, choices (A), which is ; (B), which is , which is ; and (E), which is , do not follow from those three implications.
The problems on this page are the property of the MAA's American Mathematics Competitions