AMC 10 · 2006 · #20
Grade 7 probabilitygeometry-3dPick an answer.
Because every 7-move sequence is equally likely, the probability is just (number of successful walks) divided by (number of all walks) — that is the subproblem split (Tool #7). Counting all walks is easy: 3 choices per move. The hard half is counting the successful walks, which are exactly the routes that touch all 8 corners without repeating one. To count those cleanly you must name the corners, so draw and label the cube (Tool #1) and hold its shape in mind (Tool #17). Then walk through the possibilities in an orderly way (Tool #2, the 'how many ways' tool): fix the first move by symmetry, and follow the forced branches until every good route is listed. A systematic list is the right instrument because the successful routes are few and highly constrained — most partial walks paint themselves into a corner.
Turn probability into a count of routes
Equal steps turn the probability into a count of routes.
When every outcome is equally likely, probability is just a fraction of favorable cases over total cases.
7.SP.C.7Identify SubproblemsCount all possible routes
Multiplying the choices gives 2187 routes in all.
Independent choices multiply, so 7 moves with 3 options each give 3⁷ routes.
Seven moves with three options each give three to the seventh routes in all.
▸ Why?
Each move is chosen without regard to the others, so the option counts multiply step by step.
▸ Why?
Every route is just as likely as any other, so the chance is a count of good routes over the whole count.
Label the cube and pin down the first move
Symmetry lets one first move stand for all three.
Symmetry lets you solve one representative case and scale up, instead of redoing identical work.
7.SP.C.8Draw A DiagramList every good route after A to B
Listing the forced continuations gives 6 good routes per branch.
Once two moves are fixed, the 'visit all, repeat none' rule forces the rest, so only a handful of routes survive.
7.SP.C.8Make A Systematic ListTotal the good routes and form the probability
Totalling and reducing gives 2/243, choice (C).
Good routes over all routes, reduced to lowest terms, is the answer.
4.OA.A.3Make A Systematic ListEvery 7-move path is equally likely, so the answer is just (paths that hit all 8 corners once) over (all 3⁷ = 2187 paths); careful listing shows only 18 good paths, giving 18/2187 = 2/243.
- Turn probability into a count of routes
- Count all possible routes
- Label the cube and pin down the first move
- List every good route after A to B
- Total the good routes and form the probability