AMC 10 · 2025 · #3

Grade 4 algebra

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence

pattern-recognitionsequences-geometricdigit-sum easier-related-problem ↑ Prerequisites: pattern-recognition

📏 Medium solution 💡 2 insights

Problem

A Pascal-like triangle has $10$ as the top row and $10$ followed by $1$ as the second row. In each subsequent row the first number is $10$ , the last number is $1$ , and, as in the standard Pascal Triangle, each other in the row is the sum of the two numbers directly above it. The first four rows are shown below.
$\large{10}$
$\large{10}\qquad\large{1}$
$\large{10}\qquad\large{11}\qquad\large{1}$
$\large{10}\qquad\large{21}\qquad\large{12}\qquad\large{1}$
What is the sum of the digits of the sum of the numbers in the $11$ th row?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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.

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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

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

#9 Solve an Easier Related Problem 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 $88$.

$$S_1=10,\quad S_2=11,\quad S_3=22,\quad S_4=44,\quad S_5=88$$

💡 Do the easy small cases first so a pattern can show itself.

#5 Look for a Pattern 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.

$$S_{n+1}=2S_n\ \text{for}\ n\ge 2$$

💡 Each number feeds two numbers in the row below, so the whole total copies itself.

#5 Look for a Pattern 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.)

$$S_{11}=11\times 512 = 5632$$

💡 Once you know it doubles, you never need the messy middle numbers at all.

#7 Identify Subproblems 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.

$$5+6+3+2=16\;\Rightarrow\;\textbf{(D)}$$

💡 Re-read the last line of the question so you add digits, not stop at the big number.

[1] #9 4.NBT.B.4 Total each of the small rows that are already given. Row 1 is just $10$. Row 2 i

[2] #5 4.OA.C.5 Starting from row 2, each total is exactly twice the one before: $11,\ 22,\ 44,\

[3] #5 4.NBT.B.5 Row 2 has total $11$, and going to row 11 means doubling $9$ more times. Keep do

[4] #7 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.5 Generate 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.4 Fluently 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.5 Multiply 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

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

Review

CCSS standards used (min grade 4)

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