Competition · AMC preparation · step 4 of 4
AMC 8 · 2024 · #13
Grade 2 counting
Pick an answer.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The answer choices are tiny (the biggest is 12), so the entire universe of candidate sequences is small enough to LIST. Tool #2 (Systematic List) is exactly right: pick an ordering rule and write every candidate down once. To keep track of "is Buzz underground yet?" at each step, we use Tool #1 (Draw a Diagram) — a little stair-height number line under each sequence. Finally, Tool #3 (Eliminate Possibilities) crosses out any sequence that dips below 0, leaving only the valid ones to count. We deliberately avoid Tool #13 (Algebra) or the Catalan-number formula — an elementary student does not need them when listing is fast and trustworthy.
Fix three ups and three downs
Each U is +1 stair, each D is -1, and 6 hops must return to 0 — so the ups and downs balance: exactly 3 U's and 3 D's.
Splitting 6 hops into two equal groups of 3 is a simple addition/subtraction word-problem move from Grade 1.
1.OA.A.1Draw A DiagramPin the first and last hop
First hop must be U (else underground), last must be D (to land at 0), so every sequence is U _ _ _ _ D with 2 U's and 2 D's inside.
Reasoning "if I subtract 1 from 0 I go below zero, so the first move can't be a subtraction" is everyday Grade-1 add/subtract thinking.
1.OA.A.1Draw A DiagramList the six middle arrangements
Place the two middle U's by ordered slot-pairs (1,2),(1,3),(1,4),(2,3),(2,4),(3,4), giving 6 candidate sequences.
Putting items in a fixed order so nothing is missed or repeated is the "classify and count" idea from Kindergarten.
K.MD.B.3Make A Systematic ListTrack the height of each
Trace each height from 0 (+1 for U, -1 for D); all stay at or above 0 except UDDUUD, which hits -1 after three hops — eliminate it.
Tracking a running total with +1 and -1 across several steps and checking whether it stays at or above 0 is a two-step add/subtract task from Grade 2.
Tracking Buzz's height after each hop as a running total of +1 for up and -1 for down shows that exactly one of the six candidate sequences drops below the ground, so it is the only one crossed out.
▸ Why?
His height after any hop equals the running total of his moves so far — start at 0, add 1 for each up-hop, subtract 1 for each down-hop — so every in-between height can be read straight off the sequence.
▸ Why?
Each hop contributes its own +1 or -1 to the height, with no move skipped and none counted twice, so those pieces add back to give the height at that step.
▸ Why?
Buzz goes underground exactly when a down-hop lands with no earlier up-hop left to cancel it, and among the six candidates only UDDUUD has such a down-hop — its third hop pushes the running total to -1.
▸ Why?
A down-hop reverses an up-hop, so a down-hop taken once every earlier up-hop has already been reversed must drop the height below where Buzz started.
Count the ones that survive
Five sequences survive — UUUDDD, UUDUDD, UUDDUD, UDUUDD, UDUDUD — so the answer is 5, choice (B).
After we sorted the candidates into "valid" and "invalid" groups, counting how many are in the valid group is exactly the K-grade classify-and-count skill.
K.MD.B.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 2 step-by-step adding-and-subtracting you already know!
- Fix three ups and three downs
- Pin the first and last hop
- List the six middle arrangements
- Track the height of each
- Count the ones that survive
A parent dashboard for the family lives at sensimlab.com.