AMC 10 · 2011 · #13
Grade 7 logicalgebraPick an answer.
The four numbers only matter through their gaps, so Tool #4 (Introduce a Variable) names the three neighbor gaps a=w-x, b=x-y, c=y-z. Every one of the six pairwise differences is then a run of consecutive gaps, which turns the puzzle into a couple of small equations. Adding all six differences pins down the middle gap b instantly. Tool #2 (Make a Systematic List) then sorts the leftover differences to find the outer gaps, and Tool #3 (Eliminate Possibilities) rules out the arrangements that don't give integers, leaving just the two number sets whose w values we add.
Name the three neighbor gaps
Three neighbouring gaps describe everything.
The numbers only interact through their gaps, so naming the gaps captures everything that matters.
6.EE.B.6Introduce A VariableAdd up all six differences
Adding all six pins the middle gap at 1.
The middle gap b shows up one extra time in the total, so the total minus the rest reveals it.
Adding up all six differences counts the middle gap one extra time, so the total gives it away.
▸ Why?
Each difference is built out of the gaps it spans, so the total is the gaps counted with multiplicity.
▸ Why?
The outer gaps are each covered the same number of times, so only the middle one stands out.
Find the two outer gaps
The two outer gaps must be 3 and 5.
Two gaps and each gap-plus-one must fill four slots, so the pair adding to 8 is the only fit.
7.EE.B.4Make A Systematic ListTurn each gap pattern into numbers
Each arrangement gives its own largest value.
Once the gaps are set, the fixed total 44 locks the top number to a single value.
6.EE.B.7Eliminate PossibilitiesAdd the possible values of w
Their sum is 31, choice (A).
Both gap orders produce genuine solutions, so both top values count toward the sum.
7.NS.A.1Introduce A VariableTrack the gaps between neighbors instead of the numbers themselves, and every pairwise difference becomes a short sum you can solve.
- Name the three neighbor gaps
- Add up all six differences
- Find the two outer gaps
- Turn each gap pattern into numbers
- Add the possible values of w