Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #18
Grade 4 pattern
Pick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Jumping straight to a 15-by-15 count is messy. Tool #9 (Easier Related Problem) says: try the same tiling on the smallest odd-side floors first — 1 × 1, 3 × 3, 5 × 5, 7 × 7 — and count whites by hand. Tool #5 (Look for a Pattern) then turns the four counts into a rule we can apply to side 15. The picture already does the 7 × 7 case for us, so most of the work is just counting and matching.
Count the white tiles
A tile is white only where both its row and column are odd, so the 7 × 7 has 4 odd rows times 4 odd columns = 16 whites.
Partitioning a square into rows and columns of unit tiles and counting an intersection pattern is a Grade 2 array idea.
2.G.A.2Solve An Easier Related ProblemRepeat on smaller floors
Counting the same way, 1 × 1 gives 1 white, 3 × 3 gives 4, and 5 × 5 gives 9 — joining 16 to make the list 1, 4, 9, 16.
Smaller, hand-countable versions of the same tiling let us see the structure without algebra.
2.G.A.2Solve An Easier Related ProblemSpot the square-number pattern
Those counts are the squares 1², 2², 3², 4², so a floor whose side is the k-th odd number has k² white tiles.
Generating a square-number pattern from a tiling rule is exactly the Grade 4 "generate a number or shape pattern" standard.
On the floor whose side length is the k-th odd number, the number of white tiles is k × k.
▸ Why?
A tile is white exactly when its row number and its column number are both odd, because the black stripes fill the even-numbered rows and columns, and a tile stays white only if it dodges every stripe.
▸ Why?
The floor splits with no gaps or overlaps into the tiles covered by black stripes and the tiles left over, so a tile is white precisely when it sits in none of the even-position stripes — that is, when both its row and its column are odd.
▸ Why?
Among the positions 1 through 2k-1 there are exactly k odd ones, so the floor has k odd rows and k odd columns.
▸ Why?
The odd positions 1, 3, 5, …, 2k-1 pair off one for one with the counting numbers 1, 2, 3, …, k, so the two groups are the same size and there are k odd positions.
▸ Why?
The white tiles sit at every odd-row, odd-column crossing, so they form k rows with k white tiles in each, and k equal groups of k is k × k.
Find which odd number 15 is
The odd numbers 1, 3, 5, 7, 9, 11, 13, 15 put 15 in eighth place, so k = 8.
Recognizing odd numbers as 1, 3, 5, … and locating 15 in the sequence is a Grade 3 arithmetic-pattern move.
3.OA.D.9Look For A PatternApply the pattern
Applying the rule at k = 8 gives 8² = 64 white tiles — choice (C).
Computing 8 × 8 = 64 is exactly the Grade 3 "multiply fluently within 100" standard.
3.OA.C.7Look For A PatternThis AMC 8 problem only needs Grade 4 pattern-finding with square numbers you already know!
- Count the white tiles
- Repeat on smaller floors
- Spot the square-number pattern
- Find which odd number 15 is
- Apply the pattern
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2017 #11 Gr 4Similar levelSame sub-type: Figure-N Growth SequenceSame techniqueSolve an Easier Related Problem Pattern Recognition
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals…
- AMC 10 2018B #8 Gr 4Similar level
Same sub-type: Figure-N Growth SequenceSame techniqueSolve an Easier Related Problem Pattern RecognitionSara makes a staircase out of toothpicks as shown: This is a 3-step staircase and uses 18 toothpicks. How many steps wou…
- AMC 10 2024A #6 Gr 4Similar levelSame sub-type: Figure-N Growth SequenceSame techniqueSolve an Easier Related Problem Pattern Recognition
What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the st…