Library / /359 of 387

, 2026

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 $8+3=11$ .

Figure for this problem

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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