AMC 10 · 2010 · #23
Grade 7 countingPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
"How many arrangements" is the classic trigger for Tool #2 (Make a Systematic List). Nine digits in nine cells is too many to list blindly, so first Tool #14 (Extreme Principle) pins the forced cells: the top-left must be the smallest digit and the bottom-right the largest, and the center is squeezed to just three possible values. That splits the count into a few clean cases — Tool #7 (Identify Subproblems) — organized by the center digit. Inside each case the remaining digits fall into small ordered groups that are easy to count with the multiplication principle. Tool #5 (Look for a Pattern) then reuses one case for its mirror image, halving the work.
Pin the two corners
Cell a is smaller than all eight other digits, so a is 1; cell i is larger than all eight, so i is 9.
The overall smallest digit can only sit where nothing is required to be smaller, and that is the top-left corner; the largest sits in the opposite corner.
1.NBT.B.3Extreme PrincipleTrap the center
Three cells (a, b, d) must be below the center e and three (f, h, i) above it, so e is 4, 5, or 6.
Three cells must be below the center and three above it, so the center digit cannot be too small or too large.
1.NBT.B.3Extreme PrincipleSplit by the center digit
Count the grids with center 4, with center 5, and with center 6 separately, then add the three totals.
Fixing the pivot value in the middle turns one hard count into three small independent counts.
7.SP.C.8Identify SubproblemsCount center = 4
Cells b and d take 2 and 3 in either order (2 ways), and picking which two of 5, 6, 7, 8 go in the right column fixes the rest: 12.
Once the small digits are placed, splitting the four leftover digits into an ordered pair-and-pair is just choosing two of four.
3.OA.A.3Make A Systematic ListCount center = 6 by symmetry
Swapping each digit k for 10-k and flipping the grid pairs every center-4 grid with exactly one center-6 grid, so this case also gives 12.
The rule "replace k by 10-k" is a mirror that swaps small and large, matching every center-4 grid with exactly one center-6 grid.
Replacing each digit by its partner is a mirror that swaps small and large.
▸ Why?
Each grid maps to exactly one mirrored grid, so the two families are the same size.
▸ Why?
The mirror keeps the arrangement's structure intact, so a valid grid maps to a valid grid.
Count center = 5
b and d are small, f and h are large, and one of c, g takes the last small digit; each choice allows 3 × 3 = 9, so 18.
The center-5 grid has one extra degree of freedom — which diagonal cell holds a small digit — and each choice leaves two tidy ordered triples.
3.OA.A.3Make A Systematic ListAdd the cases
Add the three case counts, 12 + 18 + 12, for the total number of valid grids: 42.
The three center values cover every grid with no overlap, so the counts simply add.
2.NBT.B.5Make A Systematic ListLock the forced cells first — 1 in the top-left, 9 in the bottom-right, and the center can only be 4, 5, or 6 — then count each center case separately: 12+18+12=42.
- Pin the two corners
- Trap the center
- Split by the center digit
- Count center = 4
- Count center = 6 by symmetry
- Count center = 5
- Add the cases