AMC 10 · 2005 · #13
Grade 6 geometry-2d
Pick an answer.
Placing 3, 5, 6, 7, 9 correctly is a maze of 120 orderings, and chasing the right one is slow. Tool #16 (Change Focus) sidesteps it: instead of finding the arrangement, add up all five edge sums at once. Tool #5 (Look for a Pattern) supplies the key structural fact — every point touches exactly two edges, so that grand total counts each number twice. Tool #4 (Introduce a Variable) names the middle term m and turns the arithmetic-sequence fact into a single equation.
Stop arranging, start totaling
Change focus from the arrangement to the grand total of all edge sums.
The middle term of an arithmetic sequence is fixed by the total of its terms, so the grand total is worth more than any single arrangement.
4.OA.A.3Change Focus Count The ComplementEach point is counted twice
Each point lies on two edges, so the total is twice the number sum.
Adding every edge sum double-counts each corner, so the total is simply twice the sum of all the numbers.
Adding up every edge sum counts each corner number twice, so the grand total is twice the sum of all the numbers.
▸ Why?
Each corner belongs to exactly two edges, so summing over edges visits every corner exactly twice.
▸ Why?
Evenly spaced numbers balance around their centre, so their average lands exactly on the middle term.
Middle term equals the average
In an odd-length arithmetic sequence the middle term is the average.
Evenly spaced numbers balance around their center, so their average lands exactly on the middle term.
6.SP.B.5Introduce A VariableSolve for the middle term
Dividing gives 12, choice (D).
One total split into five equal shares — the middle term is just the grand total divided by five.
6.EE.B.7Introduce A VariableWhen the exact arrangement is hard to pin down, add everything up at once — the total often hands you the answer.
- Stop arranging, start totaling
- Each point is counted twice
- Middle term equals the average
- Solve for the middle term