AMC 10 · 2014 · #3
Grade 2 logiccountingPick an answer.
There are only 4!=24 orderings, a small finite world, so Tool #2 (Make a Systematic List) can sweep it without missing or repeating a case. To make the rules mechanical, Tool #4 (Introduce a Variable) numbers the four spots 1,2,3,4, turning "before" into "smaller spot number" and "not next to each other" into "spot numbers differ by at least 2." The trick that keeps the list short is to place blue and yellow first, because those two colors carry two of the three rules; Tool #3 (Eliminate Possibilities) then throws out every blue-yellow placement that is adjacent, leaving only a handful to finish.
Number the four spots
Numbering the spots makes the rules concrete.
Numbering the spots turns "before" and "next to" into plain comparisons of whole numbers.
2.OA.A.1Introduce A VariablePlace blue and yellow first
The tighter pair has only three placements.
Starting with the two colors that share the most rules kills the illegal cases right away.
Starting with the two colours that share the most rules kills the illegal cases right away.
▸ Why?
An ordering rule is a comparison, and comparisons chain, so one placement rules out a whole family at once.
▸ Why?
Once the constrained colours are placed, the remaining rule leaves the last two no choice at all.
Fill in orange and red
The other two are then forced.
Once blue and yellow sit down, "orange before red" leaves no choice for the last two houses.
2.OA.A.1Make A Systematic ListCount the finished orderings
So there are 3 orderings, choice (B).
Three legal blue-yellow spots, each finishing in just one way, add up to three orderings.
2.OA.A.1Make A Systematic ListNumber the four spots, place blue and yellow first with a gap between them, and "orange before red" fills in the rest — only three rows survive.
- Number the four spots
- Place blue and yellow first
- Fill in orange and red
- Count the finished orderings