AMC 10 · 2025 · #3
Grade 4 algebraPick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Building all eleven rows just to add them is heavy, so Tool #9 (Solve an Easier Related Problem) computes the totals of the first few short rows instead. Those totals expose a clean doubling rule, which Tool #5 (Look for a Pattern) extends straight to row 11 without ever writing the giant middle rows. Tool #7 (Identify Subproblems) keeps the two-stage question tidy: first find the row's total, then separately add the digits of that total.
Add up the first few short rows
Total the short rows already drawn: 10, then 11, then 22, then 44; building row 5 gives 88.
Do the easy small cases first so a pattern can show itself.
4.NBT.B.4Solve An Easier Related ProblemSpot the doubling rule
From row 2 on, each total is exactly double the one before: 11, 22, 44, 88 — every number feeds two numbers below.
Each number feeds two numbers in the row below, so the whole total copies itself.
Each number feeds two numbers in the row below, so the whole row total doubles.
▸ Why?
Each entry is used once in each of its two children, which doubles its contribution.
▸ Why?
Multiplying by the same factor each row makes the totals a run with a fixed ratio.
Double from row 2 up to row 11
Row 2 totals 11, and reaching row 11 takes 9 more doublings: 11 times 512 = 5632.
Once you know it doubles, you never need the messy middle numbers at all.
4.NBT.B.5Look For A PatternAdd the digits of the total
The ask is the digit sum, not the total itself: 5+6+3+2 = 16, which is choice (D).
Re-read the last line of the question so you add digits, not stop at the big number.
4.NBT.B.4Identify SubproblemsOnce you notice the row totals just keep doubling, jump straight to the 11th total, then remember to add its digits.
- Add up the first few short rows
- Spot the doubling rule
- Double from row 2 up to row 11
- Add the digits of the total
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