AMC 10 · 2009 · #25

Grade 8 countinggeometry-2d

Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position

coordinate-geometryrotation-isometrysystematic-enumerationabsolute-value easier-related-problemsymmetry-argument ↑ Prerequisites: coordinate-geometryrotation-isometry

📏 Long solution 💡 4 insights 📊 Diagram

Problem

Lattice points form four separate blocks with an empty cross through the middle. Count the squares, possibly tilted, whose corners all lie in them and whose side is at least six.

Pick an answer.

(A)

125

(B)

150

(C)

175

(D)

200

(E)

225

How to solve

Strategy Solve an Easier Related Problem

Counting tilted squares in a hole-punched 10×10 board directly is hopeless, so the plan is to turn it into counting squares in a plain 5×5 grid — Tool #9 (Solve an Easier Related Problem). Getting there needs three preparatory facts. Tool #14 (Extreme Principle) measures the widest gap inside one 5×5 block and the narrowest gap between two blocks; those two numbers straddle 6, which turns 'side at least 6' into the clean condition 'one vertex in each block'. Tool #3 (Eliminate Possibilities) settles which vertices are diagonally opposite, using the fact that a square's two diagonals share a midpoint. Tool #4 (Introduce a Variable) then names a square by a corner A and a side vector w, so that sliding the three outer blocks on top of the first block becomes a single substitution w↦ w+(10,0). Tool #7 (Identify Subproblems) splits the resulting "does it fit in the 5×5 grid" test into a horizontal test and a vertical test, and Tool #2 (Make a Systematic List) sums the count over all legal side vectors.

1STEP 1

Side at least 6 means one vertex per block

The side condition forces exactly one corner per block.

same block → d ≤ 4√2 ＜ 6; different blocks → d ≥ 6

2STEP 2

Opposite corners are diagonally opposite blocks

Opposite corners must sit in diagonally opposite blocks.

midpoint(A,B)_y ≥ 3 ≠ midpoint(C,D)_y ≤ -3 → AB is not a diagonal

3STEP 3

Name the square by a corner and a side vector

One corner and one side vector name the whole square.

B=A+w, D=A+R(w), C=A+w+R(w), R(p,q)=(-q,p)

4STEP 4

Slide the three outer blocks onto the first

Sliding the outer blocks onto the first makes them comparable.

A'=A, B'=B+(10,0), C'=C+(10,10), D'=D+(0,10), w'=w+(10,0)

5STEP 5

When does a side vector fit in the grid

A side vector fits exactly when its size is small enough, giving a count.

fits ⇔ |m|+|n| ≤ 4, #{A'} = (5-|m|-|n|)²

6STEP 6

Add up over every legal side vector

Adding over every legal vector gives 225, choice (E).

Σ_k=0⁴ (ring size) (5-k)² = 25 + Σ_k=1⁴ 4k(5-k)² = 25 + 4 · 50 = 225 → (E)

Answer

225

Check the count against a piece that can be counted by hand. A square in G is axis-parallel exactly when its side vector w is horizontal, that is when n=0; the formula then gives Σ_m=-4⁴(5-|m|)²=25+2(16+9+4+1)=85. Counting axis-parallel squares directly agrees: such a square needs an x-pair and a y-pair of allowed coordinates with the same difference s, and the number of allowed pairs at difference s=6,7,…,14 is 1,2,3,4,5,4,3,2,1, giving 1+4+9+16+25+16+9+4+1=85. Two independent routes to 85 support the machinery. Note this also flags a common miscount: the upright squares are 85, not 125, so the tilted ones (140) are the majority and the answer must be well above 85. Spot-check the extremes as well. The largest term k=0 gives A' anywhere in the 5×5 grid with w=(-10,0); for A'=(5,5) that is the square (5,5),(-5,5),(-5,-5),(5,-5), valid. The smallest term k=4, (m,n)=(-4,0), forces A'=(7,7) and w=(-14,0), the single largest square (7,7),(-7,7),(-7,-7),(7,-7) — correctly counted once. A tilted sample: A=(4,3) with w=(-10,1) gives (4,3),(-6,4),(-7,-6),(3,-7), all in G, side √(101)≈10.05 ≥ 6. The total 225 matches choice (E).

💡Key takeaway

Sliding the three far blocks on top of the first one turns a messy hole-punched board into a plain 5×5 grid, and every big square becomes just a starting dot plus a short arrow — count those and you get 225.

Side at least 6 means one vertex per block
Opposite corners are diagonally opposite blocks
Name the square by a corner and a side vector
Slide the three outer blocks onto the first
When does a side vector fit in the grid
Add up over every legal side vector