AMC 8 · 2015 · #18

Grade 4 algebrapattern

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A 5 × 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.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

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.

1STEP 1

In any arithmetic sequence with an odd number of terms, the middle term equals the average of the first and last terms.

middle =

\frac{first + last}{2}

2STEP 2

Row 1 runs from 1 to 25, so its middle cell (Column 3) is $\frac{1 + 25}{2}$ = 13.

Row 1, Col 3 =

\frac{1 + 25}{2}

\frac{26}{2}

= 13

3STEP 3

Row 5 runs from 17 to 81, so its middle cell is $\frac{17 + 81}{2}$ = 49.

Row 5, Col 3 =

\frac{17 + 81}{2}

\frac{98}{2}

= 49

4STEP 4

Column 3 runs from 13 down to 49, so X = $\frac{13 + 49}{2}$ = 31 → (B).

X =

\frac{13 + 49}{2}

\frac{62}{2}

= 31 → (B)

Answer

Cross-check Column 3 with its common difference. From 13 to 49 across 5 terms means common difference d =

\frac{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

\frac{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.

💡Key takeaway

Big 5 × 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!

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