AMC 10 · 2024 · #14
Grade 8 algebraPick an answer.
The grid has 25 entries but only two "free" numbers control everything once a corner is fixed. Anchor at A₅,5 = 0: call a the common difference of column 5 going up, and b the common difference of row 5 going left. That alone uses Tool #13 (Convert to Algebra) to label the bottom row and right column with multiples of a and b. Tool #5 (Look for a Pattern) supplies the bridge: in any AP of length 5, the middle term is the mean of the endpoints. Applying that to row 3 (where A₃,1 = 12 pins one endpoint) and to row 2 (where A₂,4 = 48 is the mean of its two neighbors) gives a clean 2 × 2 linear system in a, b. Tool #7 (Identify Subproblems) keeps the work in order: first solve for a, b; then find row 1's common difference from two known row-1 entries; then read off A₁,2. We do not need a 25-variable assault — two variables suffice.
Start from the zero
The last row and column each ride a single difference.
Two letters control an entire row and an entire column — the Grade 6 "variable stands for a number" move.
6.EE.A.2Convert To AlgebraRead the middle cell along its row
Cell three-three is the average of its ends.
In a length-5 AP, the middle term is the average of the two ends — same as the Grade 6 "mean of two numbers" rule.
In an evenly stepped list of five, the middle term is the average of the two ends.
▸ Why?
Each term is the previous one plus the same fixed step, so the middle sits halfway between the ends.
▸ Why?
The halfway value is their total shared between two, which is exactly what an average is.
Read the same cell along its column
Down its column the same cell gets a second expression.
An AP is determined by any two of its terms — subtract to find the common difference, then add it to step further.
6.EE.A.2Convert To AlgebraThe first equation
Equating them gives a plus twice b equals 26.
Two routes to the same cell give an equation — a Grade 6 "solve px + q = r" setup.
6.EE.B.7Identify SubproblemsThe second equation from 48
The 48 gives three a minus four b equals 48.
Same middle-term trick, applied in row 2 — three consecutive AP terms have the middle equal to the average of the outer two.
7.EE.B.4Look For A PatternSolve the system
The two differences are 20 and 3.
Elimination on a two-by-two linear system — exactly the Grade 8 systems-of-equations skill.
8.EE.C.8Convert To AlgebraFind the first row's difference
Row one steps by 17.
Once two terms of an AP are known, one subtraction (and a divide) gives the common difference.
6.EE.B.7Identify SubproblemsRead the target cell
46 minus 17 is 29.
One subtraction inside row 1's AP finishes the problem.
6.EE.A.2Identify SubproblemsWhen a grid is made of AP rows and AP columns, two letters at one corner control everything. Use "middle term equals the mean of the endpoints" to turn the given entries into a 2 × 2 linear system, then back-fill the row you need.