AMC 10 · 2020 · #5

Grade 6 arithmetic
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 negative ten through fourteen in a five by five square so every row, every column, and both main diagonals add to the same number. Find that common sum.

Pick an answer.

(A)
2
(B)
5
(C)
10
(D)
25
(E)
50
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

Cancel the symmetric part

Opposite numbers cancel each other.

(-10) + (-9) + … + 9 + 10 = 0
2STEP 2

Add what is left

Only four numbers remain to add.

11 + 12 + 13 + 14 = 50
3STEP 3

Grand total and row sum

The grand total is five times a row sum.

Grand total = 5C = 50
4STEP 4

Divide

Divide by five.

C = 50/5 = 10
5STEP 5

Match the choice

The common sum is 10.

C = 10 → (C)
Answer
10
Cross-check by computing the average. The mean of the 25 numbers is 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 (-10 + 14)/2 = 2), so 5 · 2 = 10 is exactly what a balanced row would carry.
💡Key takeaway

This AMC 12 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.

More like this

Same archetype — closest grade level first.