AMC 10 · 2003 · #12
Grade 4 arithmeticPick an answer.
There are too many card orders to try one by one, so the smart move is to let the divisibility rule delete options until only one stack survives. Tool #2 (Make a Systematic List) first records exactly which red cards divide each blue card, turning a vague rule into a small table. Tool #14 (Extreme Principle) then spots the fussiest cards: the two reds that divide only a single blue have nowhere flexible to go and get pinned to the ends. From there Tool #3 (Eliminate Possibilities) chains each forced choice into the next until every slot is filled. Tool #1 (Draw a Diagram) keeps the nine slots in front of us so we can see the middle three.
List which red card divides each blue
Alternating nine cards puts reds at both ends; tabulate which reds divide each blue: 3:{1,3}, 4:{1,2,4}, 5:{1,5}, 6:{1,2,3}.
A red can only sit beside a blue it divides, so listing those divisors up front tells you every legal neighbor.
4.OA.B.4Make A Systematic ListPin the fussiest reds to the ends
Reds 4 and 5 fit beside only one blue each, so they are forced to the two ends.
A card that fits only one spot cannot be squeezed in the middle where it would need to fit two, so it gets pushed to a corner.
A card that fits beside only one other card cannot sit in the middle, so it is pushed to an end.
▸ Why?
A middle position needs two neighbours, so every arrangement putting that card in the middle is ruled out at once.
▸ Why?
Which cards may sit together is decided by divisibility, and each number's divisors can be listed in pairs with none missed.
Chain the forced neighbors inward
Chaining the forced neighbours inward fixes the whole pile as 5,5,1,3,3,6,2,4,4.
Each time a blue has only one red left that divides it, that neighbor is forced, and one forced choice hands you the next.
4.OA.B.4Eliminate PossibilitiesAdd the three middle cards
The middle three are 3, 3 and 6, adding to 12, choice (E).
Once the order is locked, the answer is just adding the three numbers standing in the center.
2.NBT.B.5Draw A DiagramFind the pieces that fit in only one place, lock those down first, and each forced choice will point you to the next until the whole puzzle solves itself.
- List which red card divides each blue
- Pin the fussiest reds to the ends
- Chain the forced neighbors inward
- Add the three middle cards