AMC 10 · 2010 · #17
Grade 7 countingPick an answer.
"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
Two opposite corners are forced immediately.
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.
The overall smallest digit can only sit where nothing is required to be smaller than it.
▸ Why?
Every arrow of the ordering points one way, so a value with something below it cannot be the smallest.
▸ Why?
Only one cell has nothing pointing into it, so that cell is forced to hold the smallest digit.
Trap the center
The centre is trapped in a band of three.
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
So the count splits by the centre digit.
Fixing the pivot value in the middle turns one hard count into three small independent counts.
7.SP.C.8Identify SubproblemsCount center = 4
The smallest centre gives 12 grids.
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
Symmetry gives the largest centre 12 too.
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.
4.OA.C.5Look For A PatternCount center = 5
The middle centre gives 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
Adding the cases gives 42, choice (D).
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