Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #22
Grade 6 arithmetic
Pick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The pyramid rule turns three bottom numbers into one top number, but the relationship is not obvious by inspection. Tool #9 (Easier Related Problem) says: plug in a concrete small triple, walk the pyramid up, and see what the top cell really is in terms of A, B, C. That single experiment reveals that the middle cell is counted twice, so the top equals A + 2B + C. Once that formula is in hand, Tool #6 (Guess and Check) picks the bottom cells smartly: to maximize, put the largest digit 9 in the middle (it counts twice) and the next two largest 8, 7 on the outside; to minimize, mirror the idea with 1, 2, 3.
Try a sample bottom row
Try one concrete bottom row: A = 1, B = 2, C = 3 fills upward to a top of 8.
Start small. One concrete pyramid shows the mechanism without any algebra.
5.OA.A.1Solve An Easier Related ProblemWrite the top as a sum
Reading the top as a bottom-cell sum gives Top = A + 2B + C — the middle B is counted twice.
The middle cell B feeds both second-row cells, so it shows up twice. The outer cells A and C feed only one side each.
The two second-row cells A+B and B+C add up to A + 2B + C, so the middle bottom digit B is counted twice while each outer digit is counted once.
▸ Why?
The sum (A+B) + (B+C) can be rearranged and regrouped into A + (B+B) + C, because reordering and regrouping the added terms never changes the total.
▸ Why?
Swapping the order of terms being added leaves the total unchanged, so the two copies of B can be moved next to each other.
▸ Why?
Changing which terms are grouped and added first leaves the total unchanged, so B+B can be gathered as its own group.
▸ Why?
The gathered B + B is two copies of the same amount, which is the same as two equal groups of B, namely 2B.
Maximize the top cell
To maximize, the doubled middle takes the biggest digit: B = 9 with 8, 7 outside gives top 33.
A guess like B = 8 with outer 9, 7 gives only 32, confirming that the middle should hold the largest digit.
6.EE.A.2Guess And CheckMinimize the top cell
To minimize, the middle takes the smallest digit: B = 1 with 2, 3 outside gives top 7.
A check like B = 2 with outer 1, 3 gives 1 + 4 + 3 = 8, larger than 7, so the middle should hold the smallest digit.
6.EE.A.2Guess And CheckSubtract for the difference
Subtracting the two extremes gives the requested difference: 33 - 7 = 26.
Largest minus smallest is a plain subtraction once both extremes are known.
4.NBT.B.4Guess And CheckBuild one small pyramid by hand and the rule jumps out: the top cell is A + 2B + C, so the middle digit counts twice. Put 9 in the middle for the max and 1 in the middle for the min, and the difference is 33 - 7 = 26.
- Try a sample bottom row
- Write the top as a sum
- Maximize the top cell
- Minimize the top cell
- Subtract for the difference
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