AMC 10 · 2020 · #7

Grade 6 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

sequences-arithmeticmean-median-mode-rangesystematic-enumeration identify-subproblemseasier-related-problem ↑ Prerequisites: sequences-arithmetic

📏 Short solution 💡 2 insights

Problem

Place the 25 integers from -10 through 14 in a 5 × 5 square so every row, every column, and both main diagonals add to the same number. Find that common sum.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Two clean subproblems hide inside this magic-square question. Sub-A: find the grand total of all 25 numbers. Sub-B: each row sums to the same C, and 5 rows partition the grid, so grand total = 5C. Tool #7 surfaces those two pieces. Tool #9 (easier related) lets us pair -10 with 10, -9 with 9, etc., to compute the total in one mental step instead of summing 25 values directly. Tool #3 pins the answer down to one of the five choices.

1STEP 1

Pair each number with its negative: -10 through 10 cancels to 0, leaving only 11, 12, 13, 14.

(-10) + (-9) + … + 9 + 10 = 0

2STEP 2

Add the leftover four positives: they total 50.

11 + 12 + 13 + 14 = 50

3STEP 3

The 25 numbers total 50; the 5 rows tile the grid, so the five row sums add to that total: 5C = 50.

Grand total = 5C = 50

4STEP 4

Divide both sides by 5: C = 10.

C =

\frac{50}{5}

= 10

5STEP 5

C = 10 is choice (C); the column and diagonal rules only constrain the layout, never the common sum.

C = 10 → (C)

Answer

Cross-check by computing the average. The mean of the 25 numbers is

\frac{50}{25}

= 2, and each row holds 5 numbers, so a typical row sums to 5 · 2 = 10. That matches. The numerical center of -10…14 is 2 (since

\frac{-10 + 14}{2}

= 2), so 5 · 2 = 10 is exactly what a balanced row would carry.

💡Key takeaway

This AMC 10 problem only needs Grade 6 'negatives cancel positives' plus dividing into equal groups — the 25 numbers add to 50, the 5 row sums all equal the same number C, so 5C = 50 gives C = 10.

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