AMC 10 · 2003 · #8
Grade 2 arithmeticPick an answer.
The condition is stacked: the outer ♣ acts on the inner one. So work backwards (Tool #11) — name the inner digit sum s = ♣(x) and first ask which values of s make ♣(s) = 3. To keep that search finite, pin down the range of s with a boundary check (Tool #14): a two-digit number's digits can sum to at most 9+9=18, so s lives in 1 to 18. That turns the outer condition into just two allowed inner sums, s=3 and s=12. Then the problem splits into two independent counting subproblems (Tool #7), each solved by a short systematic list (Tool #2): count two-digit numbers whose digits add to 3, and those whose digits add to 12. Add the two counts.
Name the inner sum and bound it
The inner sum of a two-digit number lies between 1 and 18.
The biggest a two-digit digit-sum can reach is 9+9=18, so the inner value can never leave 1–18.
A two-digit digit sum can never pass eighteen, so the inner value is trapped in a short range.
▸ Why?
A two-digit number has exactly two digits, each at most nine, so their total is at most nine twice over.
▸ Why?
Anything above that ceiling is impossible, so the search never has to look past it.
Find which inner sums work
Within that range only 3 and 12 have digits adding to three.
Only 3 and 12 have digits that add to 3 once you stay at or below 18.
2.OA.B.2Work BackwardsCount two-digit numbers with digit sum 3
Two-digit numbers with digit sum three number 3.
For a small target sum the tens digit can only be 1, 2, or 3, so there are just three numbers.
1.NBT.B.2Make A Systematic ListCount two-digit numbers with digit sum 12
Those with digit sum twelve number 7.
The ones digit can be at most 9, so the tens digit must be at least 3, leaving exactly the seven values 3 through 9.
1.NBT.B.2Make A Systematic ListAdd the two counts
The two cases cannot overlap, so the total is 10, choice (E).
The two digit-sum cases are separate, so the totals simply add.
2.OA.A.1Identify SubproblemsWork from the outside in: figure out what the first digit sum must be (3 or 12), then list the two-digit numbers that hit each and add the counts.
- Name the inner sum and bound it
- Find which inner sums work
- Count two-digit numbers with digit sum 3
- Count two-digit numbers with digit sum 12
- Add the two counts