AMC 10 · 2016 · #15

Grade 4 geometry-2d
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
parityparity-coloringlogical-deduction parity-coloring ↑ Prerequisites: parity
📏 Medium solution 💡 3 insights
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Problem
The digits 1 through 9 are placed in a 3×3 grid, one per square, so that any two consecutive numbers sit in squares that share an edge. The four corner squares add up to 18. Find the number in the center square.

Pick an answer.

(A)
5
(B)
6
(C)
7
(D)
8
(E)
9

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Trying to actually fill the grid is slow and there are many arrangements. Instead, color the grid like a checkerboard (Tool #1). The chain 1→2→…→9 only steps between edge-sharing squares, which always swap color, so the numbers' even/odd pattern lines up with the colors (Tool #5). Counting how many odd and even numbers there are then forces which color the corners-and-center must be (Tool #3) — and that pins the corner-plus-center total without ever building a full grid.

1STEP 1

Color the grid like a checkerboard

Shade the grid like a checkerboard so the five corner-and-center squares are dark and neighbors always differ in color.

● & ° & ● ; ° & ● & ° ; ● & ° & ● ; ●=dark (5), °=light (4)
2STEP 2

Match parity to color

Along the chain 1→2→…→9 each step flips both the color and the number's parity, so every dark square holds one parity.

1_●→2_°→3_●→4_°→…
3STEP 3

Force the odds onto the five dark squares

Five odds {1,3,5,7,9} but only four evens, so the five dark squares must be the odds, summing to 25.

1+3+5+7+9 = 25
4STEP 4

Subtract to isolate the center

The five dark squares total 25 and the four corners total 18, so the center is 25 - 18 = 7, choice (C).

center = 25 - 18 = 7 = (C)
Answer
7
A concrete grid confirms it. The rows 3 4 5 / 2 7 6 / 1 8 9 trace the chain 1→2→…→9 along edge-sharing squares, so it is legal. Its corners are 3+5+1+9=18 as required, and the center is 7 — odd, exactly as the dark-square argument demands. The corner-plus-center cells {1,3,5,7,9} sum to 25, and 25-18=7 matches.
💡Key takeaway

Color the grid like a checkerboard: consecutive numbers must hop to a different color, so the five corner-and-center squares are exactly the five odd numbers — and 25-18 leaves the center.

  • Color the grid like a checkerboard
  • Match parity to color
  • Force the odds onto the five dark squares
  • Subtract to isolate the center

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