AMC 8 · 2015 · #18

Grade 4 algebrapattern

Archetype Arithmetic Expression Decomposition

sequences-arithmeticpattern-recognitionmean-median-mode-range identify-subproblemspattern-recognition ↑ Prerequisites: sequences-arithmeticmulti-digit-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$ . Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. The square in the center is labelled $X$ as shown. What is the value of $X$ ?

$\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: A $5 \times 5$ grid hides numbers so that every row is an arithmetic sequence and every column is an arithmetic sequence. Four corners are given — top-left $1$, top-right $25$, bottom-left $17$, bottom-right $81$ — and the center cell is labelled $X$. Find $X$.

Givens: Grid is $5 \times 5$; Each of the $5$ rows is an arithmetic sequence with $5$ terms; Each of the $5$ columns is an arithmetic sequence with $5$ terms; Corners: top-left $= 1$, top-right $= 25$, bottom-left $= 17$, bottom-right $= 81$; Answer choices: (A) $21$, (B) $31$, (C) $36$, (D) $40$, (E) $42$

Unknowns: The value $X$ in the center cell (Row $3$, Column $3$)

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #5 Look for a Pattern

Filling in $25$ unknown cells at once is too much. Tool #7 (Identify Subproblems) lets us focus on only three rows/columns and ignore the rest: find the middle of Row $1$ (between $1$ and $25$), find the middle of Row $5$ (between $17$ and $81$), then find the middle of Column $3$ from those two values. Tool #5 (Look for a Pattern) gives the trick that does each subproblem in one line — in any arithmetic sequence with an odd number of terms, the middle term equals the average of the first and last terms. So we never need to compute the common difference at all.

Execute — Answer: B

#5 Look for a Pattern 4.OA.C.5 Step 1

Notice the key arithmetic-sequence pattern.
Write a short sequence like $2, 5, 8, 11, 14$: the middle term ($8$) is exactly $(2 + 14) / 2$.
The same is true for any arithmetic sequence with an odd number of terms — the middle term equals the average of the first and last terms.

$$\text{middle} = \dfrac{\text{first} + \text{last}}{2}$$

💡 Generating a few terms of an arithmetic pattern and noticing the middle = average is exactly the Grade 4 "generate and analyze patterns" standard.

#7 Identify Subproblems 4.NBT.B.4 Step 2

Subproblem 1: find the middle cell of Row $1$.
Row $1$ is an arithmetic sequence with $5$ terms, first $= 1$ and last $= 25$.
By the pattern from Step 1, the middle term (Row $1$, Column $3$) is the average of $1$ and $25$.

$$\text{Row 1, Col 3} = \dfrac{1 + 25}{2} = \dfrac{26}{2} = 13$$

💡 Adding $1 + 25 = 26$ is the Grade 4 fluent add/subtract skill for multi-digit whole numbers.

#7 Identify Subproblems 3.OA.A.2 Step 3

Subproblem 2: find the middle cell of Row $5$.
Row $5$ is also a $5$-term arithmetic sequence, first $= 17$ and last $= 81$.
Apply the same middle = average rule.

$$\text{Row 5, Col 3} = \dfrac{17 + 81}{2} = \dfrac{98}{2} = 49$$

💡 Dividing $98 \div 2 = 49$ is the Grade 3 "interpret whole-number quotients" skill — splitting $98$ into $2$ equal groups.

#7 Identify Subproblems 4.NBT.B.4 Step 4

Subproblem 3: $X$ sits in Column $3$, Row $3$.
Column $3$ is itself an arithmetic sequence with $5$ terms.
We just found its first term ($13$, from Row $1$) and its last term ($49$, from Row $5$).
Apply the middle = average rule one more time.

$$X = \dfrac{13 + 49}{2} = \dfrac{62}{2} = 31 \;\Rightarrow\; \textbf{(B)}$$

💡 The whole problem becomes three repetitions of the same Grade 4 "add two numbers and split in half" subproblem.

[1] #5 4.OA.C.5 Notice the key arithmetic-sequence pattern. Write a short sequence like $2, 5, 8

[2] #7 4.NBT.B.4 Subproblem 1: find the middle cell of Row $1$. Row $1$ is an arithmetic sequence

[3] #7 3.OA.A.2 Subproblem 2: find the middle cell of Row $5$. Row $5$ is also a $5$-term arithm

[4] #7 4.NBT.B.4 Subproblem 3: $X$ sits in Column $3$, Row $3$. Column $3$ is itself an arithmeti

Review

Reasonableness: Cross-check Column $3$ with its common difference. From $13$ to $49$ across $5$ terms means common difference $d = (49 - 13)/4 = 9$. The column then reads $13, 22, 31, 40, 49$, and the third term is indeed $31$. The value also feels right: $X$ should land roughly between the corner averages $\tfrac{1+25+17+81}{4} = 31$ — exactly $31$ in this case, because both diagonals of an "every row and every column is arithmetic" grid pass through the center.

Alternative: Tool #3 (Eliminate Possibilities) by checking common-difference consistency on each answer choice would also work but is slower. Tool #13 (Convert to Algebra) — set the four hidden corners as $a, b, c, d$ and write a system — solves it but uses heavy machinery for a problem that the middle-equals-average pattern reduces to three additions and three halvings.

CCSS standards used (min grade 4)

4.OA.C.5 Generate a number or shape pattern that follows a given rule and identify apparent features of the pattern (Recognizing that in an arithmetic sequence with an odd number of terms, the middle term equals the average of the first and last terms.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Adding endpoint pairs $1 + 25 = 26$, $17 + 81 = 98$, and $13 + 49 = 62$ to set up each midpoint calculation.)
3.OA.A.2 Interpret whole-number quotients of whole numbers (Dividing each sum by $2$ ($26 \div 2 = 13$, $98 \div 2 = 49$, $62 \div 2 = 31$) to find the middle term of each arithmetic sequence.)

⭐ Big $5 \times 5$ grid, but you only need one Grade 4 idea: the middle of an arithmetic sequence is just the average of its first and last terms — used three times!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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