AMC 8 · 2006 · #22
Grade 6 arithmeticProblem
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Three different one-digit positive integers $A$, $B$, $C$ sit in the bottom row of a small pyramid. Each cell above is the sum of the two cells it sits on top of, all the way to a single top cell. Over all valid placements, find the difference between the largest and smallest possible top numbers.
Givens: The bottom row has three cells holding three different one-digit positive integers (so they come from $\{1,2,3,4,5,6,7,8,9\}$); Second-row cells are sums of the two bottom cells beneath them; The top cell is the sum of the two second-row cells; Answer choices: (A) $16$, (B) $24$, (C) $25$, (D) $26$, (E) $35$
Unknowns: The largest possible top number; The smallest possible top number; Their difference
Understand
Restated: Three different one-digit positive integers $A$, $B$, $C$ sit in the bottom row of a small pyramid. Each cell above is the sum of the two cells it sits on top of, all the way to a single top cell. Over all valid placements, find the difference between the largest and smallest possible top numbers.
Givens: The bottom row has three cells holding three different one-digit positive integers (so they come from $\{1,2,3,4,5,6,7,8,9\}$); Second-row cells are sums of the two bottom cells beneath them; The top cell is the sum of the two second-row cells; Answer choices: (A) $16$, (B) $24$, (C) $25$, (D) $26$, (E) $35$
Plan
Primary tool: #9 Solve an Easier Related Problem
Secondary: #6 Guess and Check
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 $1, 2, 3$.
Execute — Answer: D
5.OA.A.1 Step 1 - Try a concrete bottom row to see what the top cell actually equals.
- Pick $A = 1$, $B = 2$, $C = 3$ and fill the pyramid upward.
💡 Start small. One concrete pyramid shows the mechanism without any algebra.
6.EE.A.3 Step 2 - Read the top cell as a sum of the bottom cells.
- The top $8$ came from $(A+B) + (B+C)$, which is $A + 2B + C$.
- Check: $1 + 2 \cdot 2 + 3 = 8$.
- So the formula is $\text{Top} = A + 2B + C$ for any bottom row.
💡 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.
6.EE.A.2 Step 3 - Use the formula to maximize.
- Because $B$ is multiplied by $2$, putting the biggest digit in the middle wins.
- Set $B = 9$ and place the next two biggest one-digit numbers, $8$ and $7$, on the outside.
💡 A guess like $B = 8$ with outer $9, 7$ gives only $32$, confirming that the middle should hold the largest digit.
6.EE.A.2 Step 4 - Use the same logic to minimize.
- The middle digit is doubled, so it should be as small as possible.
- Set $B = 1$ and place $2$ and $3$ on the outside.
💡 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.
4.NBT.B.4 Step 5 Subtract to get the requested difference.
💡 Largest minus smallest is a plain subtraction once both extremes are known.
5.OA.A.1 Try a concrete bottom row to see what the top cell actually equals. Pick $A = 1$ 6.EE.A.3 Read the top cell as a sum of the bottom cells. The top $8$ came from $(A+B) + ( 6.EE.A.2 Use the formula to maximize. Because $B$ is multiplied by $2$, putting the bigge 6.EE.A.2 Use the same logic to minimize. The middle digit is doubled, so it should be as 4.NBT.B.4 Subtract to get the requested difference. Review
Reasonableness: Sanity check both extremes against the choices. The largest possible top is at most $9 + 2 \cdot 9 + 9 = 36$ if repeats were allowed, so $33$ is in range. The smallest is at least $1 + 2 \cdot 1 + 1 = 4$ if repeats were allowed, so $7$ is in range. Their difference $26$ matches answer (D), and $26$ is the only choice that fits both bounds. Spot-checking another swap also confirms the formula: bottom $2, 9, 8$ gives second row $11, 17$, top $28$, matching $2 + 2 \cdot 9 + 8 = 28$.
Alternative: Tool #13 (Convert to Algebra): let the bottom row be $A, B, C$. Build the pyramid symbolically: second row $A+B$ and $B+C$, top $(A+B) + (B+C) = A + 2B + C$. Because $B$ has the largest coefficient, choose $B = 9$ and outer digits $\{7, 8\}$ for the max (top $= 33$); choose $B = 1$ and outer digits $\{2, 3\}$ for the min (top $= 7$). Difference $= 26$. Same answer (D); the easier-case + guess-and-check path just discovers the same coefficient pattern without naming variables up front.
CCSS standards used (min grade 6)
5.OA.A.1Use parentheses, brackets, or braces in numerical expressions, and evaluate them (Filling in a small concrete pyramid, $1, 2, 3 \to 3, 5 \to 8$, to discover how the top cell is built from the bottom.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Combining $(A+B) + (B+C)$ into the equivalent expression $A + 2B + C$ so the role of the middle cell is visible.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Evaluating $A + 2B + C$ with the chosen digits to compute the maximum top ($33$) and the minimum top ($7$).)4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm (Computing $33 - 7 = 26$ to get the requested difference.)
⭐ Build 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 $33 - 7 = 26$.
⭐ Build 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 $33 - 7 = 26$.