AMC 10 · 2022 · #19
Grade 8 counting
Pick an answer.
Tool #9 (Easier Problem) and #2 (Systematic List): try the question for n = 2, 3, 4 cards by listing every permutation and counting passes. The data points fit the formula 2ⁿ - n - 1 (Tool #5). Why? Tool #16 (Complement / re-cast): pass 1 collects an initial segment {1, …, k}, and an ordering uses at most two passes iff cards {1, …, k} are in increasing relative order AND cards {k+1, …, 13} are in increasing relative order — that is, the permutation is the shuffle of two increasing subsequences. There are C(13, k) such shuffles for each k, but each non-trivial labeling of cards as 'first set' vs 'second set' is unique except for the fully sorted permutation which counts in every k. Tool #3 confirms the magnitude ≈ 8000 rules out (A)-(C).
Try the smallest case
Two cards already reveal a pattern.
Grade 3 — try a tiny case to feel out the rule.
3.OA.D.9Solve An Easier Related ProblemCheck the next case
Three cards fit the same shape.
Grade 4 — enumerate by rule and tally; pattern 2ⁿ - n - 1 shows up after just two data points.
4.OA.C.5Make A Systematic ListCharacterize at most two passes
Each card joins one of two runs.
Grade 7 — recast the picking process as a structural property of the permutation.
7.SP.C.8Change Focus Count The ComplementCount at most two passes
Two choices per card gives a power of two.
Grade 7 — each subset of positions for the low cards gives exactly one valid ordering.
Each choice of positions gives exactly one valid ordering, so counting positions counts the orderings.
▸ Why?
The rule leaves no freedom once the positions are named, so the match is exact both ways.
▸ Why?
Each position is independently in or out, so counting them is counting yes-or-no choices.
Subtract the one-pass cases
Subtracting the one-pass cases gives 8178.
Grade 8 — 2¹³ from powers of 2; remove the 13 duplicates of the sorted ordering, then subtract the sorted ordering itself as one-pass.
8.EE.A.1Look For A PatternThis AMC 12 problem only needs Grade 8 powers of 2 you already know — try n=2 and n=3 by hand, spot the pattern 2ⁿ - n - 1, then plug in n = 13 to get 8192 - 14 = 8178.
More like this
Same archetype — closest grade level first.
