Competition · AMC preparation · step 4 of 4
AMC 8 · 2017 · #11
Grade 4 geometry-2dpatternPick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We do not know the side length n, and trying to attack n² directly with the value 37 is awkward. Tool #9 (Easier Related Problem) says: shrink the floor first — draw a 2 × 2, 3 × 3, 4 × 4, 5 × 5 grid (Tool #1) and count the diagonal tiles in each. Tool #5 (Look for a Pattern) then reveals the rule: odd n gives 2n - 1 diagonal tiles, even n gives 2n. Because 37 is odd, n must be odd, and 2n - 1 = 37 pinpoints n. We finish by computing n².
Try smaller floors first
Draw tiny grids and count diagonal tiles: even sides share none, odd sides share the center — giving 4, 5, 8, 9 for n = 2, 3, 4, 5.
Generating the first few cases from a rule ("count tiles on the diagonals") is exactly what Grade 4 pattern-generating practice asks for.
4.OA.C.5Solve An Easier Related ProblemFind the diagonal-tile pattern
Line up the counts: even n gives 2n, odd n gives 2n - 1 — so the total's parity matches n's parity.
Spotting that the totals split into an "even family" and an "odd family" is a Grade 4 pattern observation, no algebra needed.
The two main diagonals of an n × n tile grid cover 2n tiles when n is even and 2n - 1 tiles when n is odd, so their total is even exactly when n is even and odd exactly when n is odd.
▸ Why?
Each diagonal passes through exactly n tiles, so the two diagonals together mark 2n tile-positions before any correction for a shared tile.
▸ Why?
A diagonal runs straight from one corner to the opposite corner, entering exactly one tile in each of the n rows, which pairs the n rows one-for-one with n tiles.
▸ Why?
The two diagonals share a tile exactly when n is odd: an odd side length leaves one middle tile that lies on both diagonals, while an even side length has the diagonals cross on a line between tiles, so we subtract 1 only in the odd case.
▸ Why?
A row of n tiles has a single middle tile exactly when n cannot be split into two equal whole halves, and that failure to split evenly is what makes n odd.
▸ Why?
2n always has a factor of 2 so it is even, while 2n - 1 is one less than an even number so it is odd; the parity of the diagonal total therefore matches the parity of n.
Solve for the side length
37 is odd, so n is odd: solve 2n - 1 = 37 to get n = 19.
Recognizing 37 as odd is a Grade 2 odd/even check; then applying the pattern formula (Tool #5) backwards gives n = 19 in one short step.
2.OA.C.3Look For A PatternSquare 19 for the total
Total tiles = area of the 19 × 19 grid: (20 - 1)² = 400 - 40 + 1 = 361, choice (C).
Multiplying two two-digit numbers using place-value strategies (here, (20-1)²) is exactly the Grade 4 multi-digit multiplication standard.
4.NBT.B.5Solve An Easier Related ProblemThis AMC 8 problem only needs Grade 4 pattern-finding and two-digit multiplication you already know!
- Try smaller floors first
- Find the diagonal-tile pattern
- Solve for the side length
- Square 19 for the total
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