AMC 10 · 2019 · #10
Grade 5 counting
Pick an answer.
Tool #14 (Extreme Principle) is the crux: work out the largest number of times each city could possibly be used, add those maxima up, and compare with the number of stops 13 roads demand. The two totals turn out to be equal, so nothing is spare and every city must be used at its maximum. Tool #1 (Diagram) supplies the coordinates and road-counts that the maximum depends on. Tool #3 (Eliminate Possibilities) then kills the wrong first move and pins down which four roads are skipped. Tool #2 (Systematic List) finishes by counting the only free choices left.
Put coordinates on the map
There are seventeen roads in all.
Coordinates give every city a name, so "the city next to that one" stops being guesswork.
5.G.A.1Draw A DiagramCount the roads at each city
Count the roads meeting each city.
Adding up road-counts counts every road twice, which is a free check that none was missed.
Adding up the road counts at every city counts each road twice, which is a free check that none was missed.
▸ Why?
Each road belongs to exactly two cities, so summing over cities visits it twice.
▸ Why?
Roads at a city are used in in-and-out pairs, so an odd count always strands one road.
How often a city can be used
Each pass uses two roads.
Roads at a city are spent in in-out pairs, so an odd road-count always strands one road.
4.NBT.B.6Extreme PrincipleThe budget is exactly full
There is no slack at all.
When the most you could do equals the least you must do, every choice is already decided.
4.OA.A.3Extreme PrincipleLocate the skipped roads
The skipped roads are almost fixed.
Counting the same skipped ends two ways — by city and by road — leaves no room for a stray skipped road.
2.OA.C.3Eliminate PossibilitiesFix the starting direction
One direction cannot be the start.
A city that can be entered only once cannot satisfy three separate demands.
4.G.A.1Eliminate PossibilitiesThe rest is forced
Every remaining road is forced.
Once four skipped roads are named the budget is spent, so nothing else may be skipped.
4.G.A.1Eliminate PossibilitiesRead the route's shape
The route settles into one shape.
The two twice-visited cities are exactly the spots where the route ties a loop and comes back.
4.G.A.1Draw A DiagramCount the turns
Multiplying the remaining choices gives 4.
Two independent two-way choices multiply, they do not add.
3.OA.A.1Make A Systematic ListThis AMC 12 problem needs only Grade 5 tools: count how many times each city could possibly be used, notice that total is exactly the number of stops 13 roads require, and the whole route is forced — only the two square loops can still be turned either way, giving 2 · 2 = 4, choice (E).
- Put coordinates on the map
- Count the roads at each city
- How often can one city be used?
- The budget is exactly full
- Locate the four skipped roads
- Paula cannot start downward
- Every remaining road is forced
- Read the shape of the route
- Count the ways to turn