AMC 10 · 2011 · #8
Grade 6 patternalgebraPick an answer.
The 'every three neighbors sum to 30' rule links overlapping groups. If I write two overlapping groups as equations and compare them, a repeating pattern pops out: the sequence cycles every three terms. Once I see that cycle I can jump straight from C and the sum rule to A+H without ever finding the individual terms.
Write the sum rule with letters
Two overlapping blocks share the same two terms.
Naming the terms with letters turns the word rule into equations I can line up and compare.
6.EE.B.6Introduce A VariableCompare the overlapping groups
Comparing them makes two terms equal.
If two equal totals share a common part, the parts left over have to be equal too.
If two equal totals share a common part, the parts left over have to be equal too.
▸ Why?
Removing the same shared piece from both leaves their difference exactly as it was, namely zero.
▸ Why?
That equality ties each term to the one three steps back, so the list repeats in threes forever.
Spot the repeat-every-three pattern
So the row repeats every three terms.
A rule that ties every term to the one three steps back forces the list to loop in threes.
4.OA.C.5Look For A PatternTurn A+H into A+B and finish
The end pair sums to 25, choice (C).
Once H is really just B in disguise, the sum rule on the very first group hands me A+B directly.
6.EE.B.7Introduce A VariableWhen overlapping groups all add to the same total, the sequence repeats, so a far-away term is really a near one in disguise.
- Write the sum rule with letters
- Compare the overlapping groups
- Spot the repeat-every-three pattern
- Turn A+H into A+B and finish