2026 AMC 8 Problem 21
Attempt Problem 21 of the 2026 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2026 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
21.
Charlotte the spider is walking along a web shaped like a -pointed star, shown in the figure below. The web has outer points and inner points. Each time Charlotte reaches a point, she randomly chooses a neighboring point and moves to that point. Charlotte starts at one of the outer points and makes moves (re-visiting points is allowed). What is the probability she is now at one of the outer points of the star?
Answer: B
Small hint:
Track only outer versus inner points
Big hint:
From an inner point, half of the neighboring points are outer
Video solution:
Click to load, then click again to play
Written solution:
From an outer point, Charlotte must move to an inner point. From an inner point, of the neighboring points are outer, so the probability of moving to an outer point is . After one move she is inner. After two moves she is outer with probability and inner with probability . After the third move, only the inner case can move to outer, with probability , so the final probability is .
Problem 21 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
- 2009 AMC 8
- 2010 AMC 8
- 2011 AMC 8
- 2012 AMC 8
- 2013 AMC 8
- 2014 AMC 8
- 2015 AMC 8
- 2016 AMC 8
- 2017 AMC 8
- 2018 AMC 8
- 2019 AMC 8
- 2020 AMC 8
- 2022 AMC 8
- 2023 AMC 8
- 2024 AMC 8
- 2025 AMC 8