Competition · AMC preparation · step 4 of 4
AMC 8 · 2018 · #19
Grade 4 countingpattern
Pick an answer.
AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only 2⁴ = 16 possible bottom rows, which is small enough to list completely (Tool #2). To stay organized and to spot why the answer must be exactly half of 16, we first solve the same puzzle for a tiny pyramid with a 2-cell bottom row (Tool #9 Easier Problem) and a 3-cell bottom row, then look for the pattern (Tool #5). The pattern says: in a sign pyramid, flipping the leftmost bottom cell always flips the top sign, so exactly half of all bottom fillings give a +. That same reasoning, verified by listing, gives the answer for 4 cells without any algebra.
State the sign rule
Read the rule off the diagram: a cell is + exactly when its two cells below match, and - when they differ.
This is a Grade 4 "generate a pattern from a given rule" setup: a clear input-to-output rule we will apply over and over.
4.OA.C.5Make A Systematic ListTry a 2-cell bottom row
Warm up on a 2-cell bottom: of its 2² = 4 rows, the top is + in exactly 2 of them — half.
Shrinking the problem (Tool #9) lets a Grade 4 student check the rule by hand and see what fraction of fillings win.
4.OA.C.5Solve An Easier Related ProblemTry a 3-cell bottom row
Now the 3-cell bottom: among its 2³ = 8 rows, the top is + in exactly 4 of them — again half.
Listing in a fixed order (Tool #2) makes the count reliable; the Grade 4 pattern rule is applied row by row.
4.OA.C.5Make A Systematic ListFind the pattern
See why: flipping the leftmost bottom cell flips every cell above it up to the top, so rows pair off half-and-half.
Spotting that the rule is reversible in one spot (Grade 4 pattern reasoning) explains why the count is always exactly half.
Exactly half of all the ways to fill the four bottom cells put a + at the very top.
▸ Why?
The fillings whose top is + and the fillings whose top is - split into two equal groups that are matched to each other and together cover every possible filling.
▸ Why?
Flipping only the leftmost bottom cell always switches the sign at the very top, so it turns each top-+ filling into a top-- filling and turns it back if done again.
▸ Why?
The switch of the leftmost cell changes the cell directly above it, and that change is passed upward one level at a time until it reaches and flips the top.
▸ Why?
A cell shows + when an even number of its two lower cells are -, and - when an odd number are -, so changing just one of those cells adds or removes exactly one -, flipping even to odd and moving the cell to its opposite sign.
▸ Why?
Flipping the leftmost cell pairs each top-+ filling with exactly one top-- filling and leaves none unpaired, so the two groups hold the same number of fillings.
▸ Why?
The top-+ fillings and top-- fillings never overlap and together are all the fillings, so two equal groups that make up the whole are each exactly half.
Apply it to 4 cells
Apply it to 4 cells: half of 2⁴ = 16 rows win, so the count is 16 ÷ 2 = 8 — choice (C).
A Grade 4 student can both apply the half-rule and verify by listing the 8 winners explicitly.
4.OA.C.5Make A Systematic ListThis AMC 8 problem only needs Grade 4 "follow a rule and look for the pattern" thinking you already know!
- State the sign rule
- Try a 2-cell bottom row
- Try a 3-cell bottom row
- Find the pattern
- Apply it to 4 cells
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