AMC 8 · 2008 · #2

Grade 4 arithmetic

Archetype Small-Domain Constrained Search

place-valuesystematic-enumerationlogical-deduction systematic-enumeration ↑ Prerequisites: place-valuemulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

The ten-letter code $\text{BEST OF LUCK}$ represents the ten digits $0-9$ , in order. What 4-digit number is represented by the code word $\text{CLUE}$ ?

Pick an answer.

(A)

8671

(B)

8672

(C)

9781

(D)

9782

(E)

9872

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Toolkit + CCSS Solution

Understand

Restated: The ten letters of the phrase $\text{BEST OF LUCK}$ stand for the digits $0$ through $9$, in the order they appear. Use that code to find which $4$-digit number is spelled by $\text{CLUE}$.

Givens: The code phrase is $\text{BEST OF LUCK}$ — exactly $10$ distinct letters; The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are assigned in order to those $10$ letters; The target code word is $\text{CLUE}$; Answer choices: (A) $8671$, (B) $8672$, (C) $9781$, (D) $9782$, (E) $9872$

Unknowns: The $4$-digit number that the letters $\text{C}, \text{L}, \text{U}, \text{E}$ encode

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #16 Use Structure / Transform

Tool #2 (Make a Systematic List) is the natural fit: line the $10$ letters up against the digits $0$–$9$ in a two-row table so every code letter has its digit right below it. Once that lookup table exists, decoding any code word is just reading off four digits. Tool #16 (Use Structure / Transform) is the underlying idea — we are turning a word into a number by applying a fixed letter-to-digit transformation, so the structure of the phrase $\text{BEST OF LUCK}$ controls the answer.

Execute — Answer: A

#2 Make a Systematic List 2.NBT.A.3 Step 1

Build the lookup table.
Write the $10$ letters of $\text{BEST OF LUCK}$ in a row, then write $0, 1, 2, \ldots, 9$ underneath them in order.
Each letter sits directly above its digit, so the table can be read either way.

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Letter} & \text{B} & \text{E} & \text{S} & \text{T} & \text{O} & \text{F} & \text{L} & \text{U} & \text{C} & \text{K} \\ \hline \text{Digit} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \end{array}$$

💡 Writing the digits $0$ through $9$ in order and matching each to one slot is exactly the Grade 2 "read and write numbers up to $1000$" skill of using place-value names in a fixed order.

#2 Make a Systematic List 2.NBT.A.3 Step 2

Look up each letter of $\text{CLUE}$ in the table.
Find $\text{C}$, $\text{L}$, $\text{U}$, $\text{E}$ one by one and copy the digit underneath.

$$\text{C} \to 8, \quad \text{L} \to 6, \quad \text{U} \to 7, \quad \text{E} \to 1$$

💡 The lookup table makes decoding pure pattern-matching: see the letter, copy the digit. No arithmetic needed.

#16 Use Structure / Transform 4.NBT.A.2 Step 3

Write the four digits in the same order as the letters in $\text{CLUE}$.
The first letter gives the thousands digit, the second the hundreds, then tens, then ones.

$$\text{C}\,\text{L}\,\text{U}\,\text{E} \to 8\,6\,7\,1 = 8671 \;\Rightarrow\; \textbf{(A)}$$

💡 Reading a $4$-digit number is just stacking thousand-hundred-ten-one place values in order — the same structure as reading $\text{CLUE}$ left to right.

[1] #2 2.NBT.A.3 Build the lookup table. Write the $10$ letters of $\text{BEST OF LUCK}$ in a row

[2] #2 2.NBT.A.3 Look up each letter of $\text{CLUE}$ in the table. Find $\text{C}$, $\text{L}$,

[3] #16 4.NBT.A.2 Write the four digits in the same order as the letters in $\text{CLUE}$. The fir

Review

Reasonableness: Cross-check against the answer choices. The first letter of $\text{CLUE}$ is $\text{C}$, the ninth letter of $\text{BEST OF LUCK}$, so the number must start with $8$. That kills (C), (D), and (E), which all start with $9$. The third letter $\text{U}$ is the eighth letter, giving a $7$ in the tens place — so the number ends in $7\_$, and the last letter $\text{E}$ is the second letter, giving $1$. The number is $86\underline{7}\,\underline{1}$, matching only (A) $8671$.

Alternative: Tool #5 (Look for a Pattern): use the position-of-letter rule directly. Number the letters of $\text{BEST OF LUCK}$ as $1, 2, 3, \ldots, 10$, then a letter at position $p$ codes the digit $p - 1$. $\text{C}$ is at position $9$, so $\text{C} = 8$; $\text{L}$ is at position $7$, so $\text{L} = 6$; $\text{U}$ is at position $8$, so $\text{U} = 7$; $\text{E}$ is at position $2$, so $\text{E} = 1$. Reading in order gives $8671$, answer (A).

CCSS standards used (min grade 4)

2.NBT.A.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form (Lining the digits $0$–$9$ up in order under the $10$ letters and reading single digits out of that ordered list.)
4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form (Combining the four decoded digits $8, 6, 7, 1$ into the $4$-digit number $8671$ in the correct place-value order.)

⭐ When a problem hands you a code, the fastest move is to write the matching chart — once the letters and digits are lined up, decoding is just reading.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 4)

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