AMC 10 · 2012 · #5
Grade 2 logicnumber-theoryPick an answer.
The question asks for a minimum, so Tool #14 (Extreme Principle) sets the shape of the work: find a floor no set can go below, then produce one set that sits exactly on the floor. Tool #7 (Identify Subproblems) makes the floor findable — the running totals cut the six integers into three independent pairs, each with its own sum. Tool #16 (Change Focus) drops the actual sizes and keeps only even-or-odd, which is all the count depends on. Tool #6 (Guess and Check) then builds a concrete list of six integers that hits every stated total while carrying as few even numbers as the floor allows.
Cut the six into three pairs
The running totals split the six into three pairs.
Two running totals differ by exactly what was added in between, so subtracting them isolates each new pair.
2.OA.A.1Identify SubproblemsWatch parity, not size
Only each pair's parity matters.
Even and odd combine in a fixed pattern, so a pair's sum reports back how many of its two numbers are even.
Even and odd combine in a fixed pattern, so a pair's sum reports how many of its two numbers are even.
▸ Why?
Two numbers of the same parity add to an even total, while a mismatched pair always lands on odd.
▸ Why?
Two running totals differ by exactly what was added in between, so subtracting isolates each new pair.
Prove zero evens is impossible
One odd pair sum forces at least one even.
A single odd pair-sum forces an even number into the list, and it forces only one.
2.OA.C.3Extreme PrincipleBuild a list that reaches one
A real list reaches 1, choice (A).
A bound you can actually land on is the true minimum; a bound nothing reaches is only a guess.
2.NBT.B.5Guess And CheckTwo odd numbers always add up to an even number, so any pair that must add to an odd total is forced to hide exactly one even number inside it.
- Cut the six into three pairs
- Watch parity, not size
- Prove zero evens is impossible
- Build a list that reaches one