AMC 10 · 2025 · #23
Grade 7 number-theoryA rectangular grid of squares has 141 rows and 91 columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from 1 through 141×91=12,831 into the squares. Horace fills the grid horizontally: he puts 1 through 91 in order from left to right into row 1, puts 92 through 182 into row 2 in order from left to right, and continues similarly through row 141. Vera fills the grid vertically: she puts 1 through 141 in order from top to bottom into column 1, then 142 through 282 into column 2 in order from top to bottom, and continues similarly through column 91. How many squares get two copies of the same number?
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 grid has $141$ rows and $91$ columns. Horace writes $1$ through $12{,}831$ row by row (left to right, top to bottom); Vera writes the same numbers column by column (top to bottom, left to right). A square gets "two copies of the same number" when Horace's number and Vera's number land in that same square. Count how many squares this happens in.
Givens: The grid has $141$ rows and $91$ columns, so $141\times 91 = 12{,}831$ squares; Horace fills horizontally: row $1$ gets $1$–$91$, row $2$ gets $92$–$182$, and so on ($91$ numbers per row); Vera fills vertically: column $1$ gets $1$–$141$, column $2$ gets $142$–$282$, and so on ($141$ numbers per column); Answer choices: (A) $7$, (B) $10$, (C) $11$, (D) $12$, (E) $19$
Unknowns: The number of squares where Horace's number equals Vera's number
Understand
Restated: A grid has $141$ rows and $91$ columns. Horace writes $1$ through $12{,}831$ row by row (left to right, top to bottom); Vera writes the same numbers column by column (top to bottom, left to right). A square gets "two copies of the same number" when Horace's number and Vera's number land in that same square. Count how many squares this happens in.
Givens: The grid has $141$ rows and $91$ columns, so $141\times 91 = 12{,}831$ squares; Horace fills horizontally: row $1$ gets $1$–$91$, row $2$ gets $92$–$182$, and so on ($91$ numbers per row); Vera fills vertically: column $1$ gets $1$–$141$, column $2$ gets $142$–$282$, and so on ($141$ numbers per column); Answer choices: (A) $7$, (B) $10$, (C) $11$, (D) $12$, (E) $19$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #5 Look for a Pattern, #2 Make a Systematic List
Comparing two whole grids of $12{,}831$ numbers is hopeless to do square by square, so Tool #4 (Introduce a Variable) names a single square by its row $r$ and column $c$ and asks only about that one square. Tool #13 (Convert to Algebra) writes what Horace wrote there and what Vera wrote there as two formulas in $r$ and $c$; a "same number" square is exactly where the two formulas are equal, which turns the whole puzzle into one equation. Tool #5 (Look for a Pattern) finds that once you have one matching square, the next one is a fixed step away — right $9$ columns and down $14$ rows. Tool #2 (Make a Systematic List) then just walks that step from the first match to the last and counts how many fit on the grid.
Execute — Answer: C
6.EE.B.6 Step 1 Name one square by row and column
- Instead of looking at all $12{,}831$ numbers, focus on a single square and label it by its row $r$ (counting $1$ at the top) and its column $c$ (counting $1$ at the left).
- Every square has such a pair $(r,c)$ with $1\le r\le 141$ and $1\le c\le 91$.
- The goal is to find the pairs where the two people wrote the same number.
💡 Giving the square an address $(r,c)$ lets you talk about any square with one formula instead of a huge list.
6.EE.A.2 Step 2 Write Horace's number there
- Horace fills $91$ numbers per row.
- To reach row $r$, column $c$, he first fills the $r-1$ complete rows above it, which uses $91(r-1)$ numbers, and then counts $c$ more squares into row $r$.
- So the number Horace writes in square $(r,c)$ is $91(r-1)+c$.
💡 Full rows before you, plus the steps into your own row, tells you exactly how far along Horace's count you are.
6.EE.A.2 Step 3 Write Vera's number there
- Vera fills $141$ numbers per column.
- To reach the same square $(r,c)$, she first fills the $c-1$ complete columns to its left, using $141(c-1)$ numbers, then counts $r$ more squares down column $c$.
- So the number Vera writes in square $(r,c)$ is $141(c-1)+r$.
💡 Same square, but now full columns to the left plus steps down the column measure Vera's count.
7.EE.B.4 Step 4 Set the two numbers equal
- A square holds "two copies of the same number" exactly when Horace's number equals Vera's number.
- Set $H=V$ and simplify.
- Expanding gives $91r-91+c = 141c-141+r$.
- Collect the $r$ and $c$ terms: $90r = 140c - 50$.
- Dividing everything by $10$ gives the clean relationship $9r + 5 = 14c$.
💡 One tidy equation replaces the whole grid: any $(r,c)$ solving it is a matching square.
6.NS.B.4 Step 5 Find the first match and the repeating step
- The top-left square $(r,c)=(1,1)$ works, since $9(1)+5 = 14 = 14(1)$: both people write $1$ there.
- To find the next solution, keep the equation balanced.
- Raising $r$ by $14$ adds $126$ to the left side, and raising $c$ by $9$ adds $126$ to the right side, so both sides stay equal.
- The step sizes are forced: because $9$ and $14$ share no common factor (their GCD is $1$), the smallest whole-number step that keeps $9r$ and $14c$ in step is $r\!\uparrow\!14$, $c\!\uparrow\!9$.
- So matches march along in a straight line, each one $14$ rows down and $9$ columns right of the last.
💡 Because $9$ and $14$ have no shared factor, the coordinates can only stay balanced by jumping a full $14$ and $9$ at a time.
4.OA.B.4 Step 6 List the matches and count them
- Start at $(1,1)$ and repeatedly add $14$ to the row and $9$ to the column: $(1,1), (15,10), (29,19), \dots$.
- The row values are $1, 15, 29, \dots, 141$ and the column values are $1, 10, 19, \dots, 91$.
- Both hit their maximum at the same last square, the bottom-right corner $(141,91)$, since $9(141)+5 = 1274 = 14(91)$.
- The row values are $r = 1 + 14k$ for $k = 0,1,2,\dots,10$, which is $11$ values, and each stays within the grid.
- So there are $11$ squares with two copies of the same number, which is choice (C).
💡 The matches are evenly spaced dots from one corner to the opposite one, so you just count the stops.
6.EE.B.6 Instead of looking at all $12{,}831$ numbers, focus on a single square and label 6.EE.A.2 Horace fills $91$ numbers per row. To reach row $r$, column $c$, he first fills 6.EE.A.2 Vera fills $141$ numbers per column. To reach the same square $(r,c)$, she first 7.EE.B.4 A square holds "two copies of the same number" exactly when Horace's number equa 6.NS.B.4 The top-left square $(r,c)=(1,1)$ works, since $9(1)+5 = 14 = 14(1)$: both peopl 4.OA.B.4 Start at $(1,1)$ and repeatedly add $14$ to the row and $9$ to the column: $(1,1 Review
Reasonableness: The first match is the top-left corner $(1,1)$ and the last is the bottom-right corner $(141,91)$ — a satisfying picture, since both people obviously write $1$ in the top-left and $12{,}831$ in the bottom-right. Between them the matches step evenly by $14$ rows and $9$ columns. Check the count another way: the row jumps from $1$ to $141$, a total climb of $140$, in steps of $14$, giving $140/14 = 10$ steps, hence $10+1 = 11$ squares; the column jumps from $1$ to $91$, a climb of $90$ in steps of $9$, giving $90/9 = 10$ steps, the same $11$. Both coordinates agree, so $11$ is solid and matches (C).
Alternative: Number the rows and columns starting from $0$ instead of $1$. Then Horace's number is $91r+c+1$ and Vera's is $141c+r+1$; setting them equal cancels the $+1$ and gives $90r = 140c$, or $9r = 14c$. Now the matches are just the points where $r$ is a multiple of $14$ and $c$ the matching multiple of $9$: $r = 0,14,28,\dots,140$, which is $11$ values. Same count, reached with slightly cleaner numbers.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Labeling an arbitrary square by its row $r$ and column $c$ so one formula can stand for any square.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Writing Horace's number $91(r-1)+c$ and Vera's number $141(c-1)+r$ for the square $(r,c)$.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Setting the two expressions equal and simplifying $91(r-1)+c=141(c-1)+r$ down to $9r+5=14c$.)6.NS.B.4Find greatest common factor and least common multiple of two numbers (Using that $\gcd(9,14)=1$ to fix the repeating step at $14$ rows and $9$ columns between matches.)4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Listing the evenly spaced solutions $r=1+14k$ and counting the $11$ that fit on the grid.)
⭐ Label a square by row and column, write each person's number as a formula, set them equal, and the matching squares turn out to be evenly spaced dots marching corner to corner — just count the stops to get $11$.
⭐ Label a square by row and column, write each person's number as a formula, set them equal, and the matching squares turn out to be evenly spaced dots marching corner to corner — just count the stops to get $11$.
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