AMC 10 · 2016 · #12
Grade 4 geometry-2dPick an answer.
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.
Color the grid like a checkerboard
A checkerboard colouring splits the grid five to four.
On a checkerboard every step to a neighbor flips the color, so corners and center share one color and the edges share the other.
On a checkerboard every step to a neighbour flips the colour, so corners and centre share one colour.
▸ Why?
Neighbouring squares always carry opposite colours, so the colouring alternates across the whole grid.
▸ Why?
Two things that both flip at every single step stay perfectly matched the whole way.
Match parity to color
Consecutive numbers must change colour.
Two things that both flip at every single step stay perfectly matched the whole way.
2.OA.C.3Look For A PatternForce the odds onto the five dark squares
So all the odd numbers sit on one colour.
Five same-parity squares can't be filled by only four evens, so they must be the odds.
2.NBT.B.5Eliminate PossibilitiesSubtract to isolate the center
Subtracting the corners gives 7, choice (C).
Corners and center together are known, and the corners alone are known, so the leftover is the center.
2.NBT.B.5Eliminate PossibilitiesColor 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