AMC 10 · 2006 · #9
Grade 7 countingPick an answer.
A "how many" question with a small finite universe calls for Tool #2 (Make a Systematic List), but listing all 450 even three-digit numbers is not the plan — the plan is to list the right objects. Tool #3 (Eliminate Possibilities) first shrinks the alphabet: no digit is 0, and the units digit is one of only three values. Then Tool #15 (Organize Information in More Ways) does the load-bearing work. A strictly increasing number carries no information beyond which three digits it uses, because their order is already forced, so a number can be re-seen as a set of digits. That re-seeing is the whole problem, and it must be argued in both directions: every valid number gives a digit set, and — the part that is usually asserted rather than shown — every allowed digit set gives back a valid number, exactly one of them. The second direction is where "no digit is 0" earns its keep, because it is what guarantees the rebuilt number really has three digits. Tool #7 (Identify Subproblems) then splits the count by units digit into three non-overlapping cases and adds them.
Rule zero out of every slot
Strictly increasing digits push zero out of every slot.
In an increasing number the smallest digit leads, so a zero anywhere would have to sit in front — and then the number would not be three digits.
2.NBT.A.1Eliminate PossibilitiesPin down the units digit
The last digit must be even and leave room, so only three remain.
Even is a fact about the last digit only, and here the last digit is also the biggest, so it has to leave two smaller digits underneath it.
2.OA.C.3Eliminate PossibilitiesMatch numbers with digit pairs
Order being fixed, each number matches a pair of smaller digits.
Once digits are forced to increase, the order carries no information — the number is nothing more than the set of digits it uses.
Once the digits are forced to increase, the order carries no information and the number is just its set of digits.
▸ Why?
Each set of digits can be arranged in increasing order in exactly one way, so sets and numbers pair up perfectly.
▸ Why?
Naming a pair in order and then halving corrects for the fact that a pair does not remember which was named first.
Count the pairs below each c
Counting the pairs below each gives 3, 10 and 21.
Choose two by naming them in order and then halving, because a pair does not remember which one you named first.
7.SP.C.8Make A Systematic ListAdd the three cases
Adding gives 34, choice (B).
The units digit tags every number with exactly one case, so no number is missed and none is counted twice.
4.OA.A.3Identify SubproblemsWhen digits are forced to increase, the order is already decided — so counting these numbers is just counting which digits to use.
- Rule zero out of every slot
- Pin down the units digit
- Match numbers with digit pairs
- Count the pairs below each c
- Add the three cases