This week’s puzzle is an oldie but a goodie, and comes from high school student Meryl Zhang of Plano, Texas, who was named the top math awardee at this year’s Regeneron International Science and Engineering Fair. For her project, Meryl developed novel heuristics for the NP-hard partition problem. Meryl enjoys K-dramas, including Squid Game—a show that has already proven ripe for riddles.
This week, the final 20 players in the Squid Game competition form a circle and are assigned the whole numbers from 1 to 20, progressing in a clockwise direction. First, contestant 2 is eliminated. Then, the contestant two positions clockwise from where 2 was (i.e., contestant 4) is eliminated. Next, the contestant two positions clockwise from them (i.e., contestant 6) is eliminated. The counting continues in this manner, wrapping around the circle, which tightens after each elimination. So after contestant 20 is eliminated, the next contestant to go is 3, who at this point is two positions clockwise from where 20 once stood.
You repeat this process until only one contestant remains as the ultimate winner of the game. What is the winner’s number?
Originally, the table is,
$$1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - 17 - 18 - 19 - 20$$
After going around the circle once and eliminating, the table is,
$$1 - 3 - 5 - 7 - 9 - 11 - 13 - 15 - 17 - 19$$
After going around the circle twice and eliminating, the table is,
$$1 - 5 - 9 - 13 - 17$$
After going around the circle three times and eliminating, the table is,
$$1 - 9 - 17$$
After going around the circle four times and eliminating, the table is,
$$9 - 17$$
After going around the circle five times and eliminating, the table is,
$$17$$
Player 17 is the winner.