AMC 10 · 2025 · #3
Easy mode Grade 4A number triangle is built one row at a time. The top row is just 10. Every row after that starts with 10 and ends with 1. Each number in between is the sum of the two numbers directly above it. Here are the first four rows.
10
101
10111
1021121
Add up all the numbers in the 11th row. Then add up the digits of that total. What do you get?
Pick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A triangle is built like Pascal's Triangle, but every row starts with $10$ and ends with $1$; each middle number is the sum of the two numbers directly above it. Find the sum of the numbers in the $11$th row, then add up the digits of that sum.
Givens: Row 1 is $10$; row 2 is $10,\ 1$; row 3 is $10,\ 11,\ 1$; row 4 is $10,\ 21,\ 12,\ 1$; In every row the first number is $10$ and the last number is $1$; Every other number is the sum of the two numbers directly above it; Answer choices: (A) $11$, (B) $13$, (C) $14$, (D) $16$, (E) $17$
Unknowns: The sum of the digits of the total of all numbers in the $11$th row
Understand
Restated: A triangle is built like Pascal's Triangle, but every row starts with $10$ and ends with $1$; each middle number is the sum of the two numbers directly above it. Find the sum of the numbers in the $11$th row, then add up the digits of that sum.
Givens: Row 1 is $10$; row 2 is $10,\ 1$; row 3 is $10,\ 11,\ 1$; row 4 is $10,\ 21,\ 12,\ 1$; In every row the first number is $10$ and the last number is $1$; Every other number is the sum of the two numbers directly above it; Answer choices: (A) $11$, (B) $13$, (C) $14$, (D) $16$, (E) $17$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #9 Solve an Easier Related Problem, #7 Identify Subproblems
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.
Execute — Answer: D
4.NBT.B.4 Step 1 Add up the first few short rows
- Total each of the small rows that are already given.
- Row 1 is just $10$.
- Row 2 is $10+1=11$.
- Row 3 is $10+11+1=22$.
- Row 4 is $10+21+12+1=44$.
- Building one more row, row 5 is $10,\ 31,\ 33,\ 13,\ 1$, which totals $88$.
💡 Do the easy small cases first so a pattern can show itself.
4.OA.C.5 Step 2 Spot the doubling rule
- Starting from row 2, each total is exactly twice the one before: $11,\ 22,\ 44,\ 88$.
- This happens because when you build the next row, every number in the current row gets used twice in the middle sums, and the fixed $10$ and $1$ you place on the two ends exactly make up for the two end numbers that were only used once.
- So from row 2 onward the total doubles every step.
💡 Each number feeds two numbers in the row below, so the whole total copies itself.
4.NBT.B.5 Step 3 Double from row 2 up to row 11
- Row 2 has total $11$, and going to row 11 means doubling $9$ more times.
- Keep doubling: $11,\ 22,\ 44,\ 88,\ 176,\ 352,\ 704,\ 1408,\ 2816,\ 5632$.
- The last value, $5632$, is the total of the $11$th row.
- (This is the same as $11$ times $512$, where $512$ is $2$ doubled onto itself nine times.)
💡 Once you know it doubles, you never need the messy middle numbers at all.
4.NBT.B.4 Step 4 Add the digits of the total
- The question asks for the sum of the digits of the row total, not the total itself.
- Add the digits of $5632$: $5+6+3+2=16$.
- That matches choice (D).
- Watch the trap of stopping at $5632$ or of doubling only $10$ times instead of $9$ from row 2.
💡 Re-read the last line of the question so you add digits, not stop at the big number.
4.NBT.B.4 Total each of the small rows that are already given. Row 1 is just $10$. Row 2 i 4.OA.C.5 Starting from row 2, each total is exactly twice the one before: $11,\ 22,\ 44,\ 4.NBT.B.5 Row 2 has total $11$, and going to row 11 means doubling $9$ more times. Keep do 4.NBT.B.4 The question asks for the sum of the digits of the row total, not the total itse Review
Reasonableness: The doubling rule can be checked directly against the given rows: $11\to22\to44$ matches rows 2, 3, 4 exactly, so trusting it for the later rows is safe. Counting the doublings is the only place to slip: row 2 is the $11$, and rows 3 through 11 are nine more doublings, landing on $5632$. Its digits add to $16$, which is one of the listed choices, and every wrong choice ($11,13,14,17$) fails to be the digit sum of $5632$.
Alternative: Work modulo $9$: a number and its digit sum leave the same remainder when divided by $9$. Since $10\equiv1\pmod 9$, the triangle behaves like ordinary Pascal's Triangle mod $9$, whose $11$th row sums to $2^{10}=1024\equiv7\pmod 9$. Among the choices only $16$ leaves remainder $7$ when divided by $9$, so the answer must be (D).
CCSS standards used (min grade 4)
4.OA.C.5Generate a number or shape pattern following a given rule (Recognizing that each row total is double the previous one and extending that rule from the given rows out to row 11.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Totaling the entries of the first few rows and adding the digits $5+6+3+2$ of the final total.)4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Doubling each row total step by step (multiplying by $2$) all the way up to $5632$.)
⭐ Once you notice the row totals just keep doubling, jump straight to the 11th total, then remember to add its digits.
⭐ Once you notice the row totals just keep doubling, jump straight to the 11th total, then remember to add its digits.
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