AMC 10 · 2012 · #23
Grade 10 geometry-2dPick an answer.
A square wandering over a 2012 × 2012 board is hopeless to track directly, so Tool #16 (Change Focus) flips the roles: nail a copy of S to every lattice point and ask how many of those fixed copies cover the random point. The count is then a fixed map of the plane. Two facts have to be nailed down before any area is computed, and both are easy to skip. Tool #14 (Extreme Principle) supplies the first: the largest possible distance between two interior points of a unit square is strictly under √(2), which forces any two captured lattice points to be orthogonal neighbours. Tool #3 (Eliminate Possibilities) supplies the second: three lattice points can never be captured, which is what makes 'exactly two' the same event as 'some neighbouring pair is captured' and — the part usually left unsaid — makes the regions belonging to different pairs disjoint, so their areas may simply be added. Tool #9 (Solve an Easier Related Problem) then shrinks the board to a single unit cell using periodicity, and Tool #7 (Identify Subproblems) reduces everything to one measurable object: the overlap of two neighbouring copies of S. Tool #1 (Draw a Diagram) underlies the whole thing by first identifying S as a tilted unit square.
Pin down the square
The diagonal fixes the square's size and tilt.
A diagonal of length √(2) is exactly the diagonal of a unit square, so S is an ordinary unit square that has been tipped over.
8.G.B.8Draw A DiagramLet the point pick the squares
Containing a point becomes a condition on the centre.
Nailing a copy of the square to every lattice point turns a moving target into a fixed map that can just be measured.
10.S-CP.A.1Change Focus Count The ComplementTwo inside means side by side
Two points inside forces them to be neighbours.
The square is just barely too small to hold both ends of a diagonal step, and every other lattice step is far too long.
8.G.B.8Extreme PrincipleThree is impossible, so nothing overlaps
Three inside is impossible, so nothing overlaps.
Because one copy can never swallow three lattice points, no center is ever claimed by two different pairs, so the pieces never double up.
Because one copy can never hold three lattice points, no centre is ever claimed by two different pairs.
▸ Why?
Each centre belongs to at most one pair, so the regions never overlap and their areas simply add.
▸ Why?
Each good centre is matched with exactly one pair of points, so counting centres counts the pairs once each.
One unit cell is the whole story
One unit cell is the whole story.
The pattern of good centers tiles the plane and the board is a whole number of tiles, so one tile decides everything.
7.SP.C.7Solve An Easier Related ProblemMeasure one overlap
Each overlap is a small rectangle.
A tilted square is two perpendicular strips crossed, and sliding it sideways just trims each strip by how far the slide pushes across it.
10.G-GPE.B.4Identify SubproblemsAdd the two directions
Adding both directions gives 4/25, choice (C).
Two pair directions, one overlap rectangle each, and the two rectangles never collide — so the areas just add.
10.G-GPE.B.7Identify SubproblemsNail a copy of the tilted square to every grid point: it is just wide enough to cover two side-by-side grid points at once but never two diagonal ones, so the answer is simply how much of a cell two neighbouring copies share.
- Pin down the square
- Let the point pick the squares
- Two inside means side by side
- Three is impossible, so nothing overlaps
- One unit cell is the whole story
- Measure one overlap
- Add the two directions
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