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Number theory Difficulty 3.4 AMC 10/12 Find the answer Canada

A Nelson card has a number on each side. Three Nelson cards lie
on a table, as shown.

Jen flips the cards one by one, flipping the $3$ , then the $7$ , and then the $8$ . Each time she flips a card, the mean
(average) of the three visible numbers increases by $1$ . What number is on the opposite side of
the card with the $8$ ?

Pick one

Solution

Initially, the mean of the three visible numbers is $\dfrac{3+7+8}{3}=\dfrac{18}{3}=6$ . After
the $3$ is flipped, the new mean is
$7$ .

If the number on the other side of the card with the $3$ is $x$ , then $\dfrac{x+7+8}{3}=7$ or $x+15=3\times7$ and so $x=21-15=6$ . Note that the mean of the
numbers on the three cards increased by $1$ when the $3$ was flipped, and so it follows that the
sum of the three visible numbers increased by $3$ . So, the new card (the opposite side of
the card with the $3$ ) must be $3+3=6$ .

Similarly, since the mean increased by $1$ again when the $7$ was flipped, the number on the opposite
side of the card with the $7$ is
$7+3=10$ .

Finally, the mean increased by $1$
when the $8$ was flipped and so the
number on the opposite side of the card with the $8$ is $8+3=11$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.