AMC 8 · 2008 · #2
Grade 4 arithmeticProblem
The ten-letter code BEST OF LUCK represents the ten digits 0−9, in order. What 4-digit number is represented by the code word CLUE?
Pick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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) $9872$
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.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.
💡 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.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.
💡 The lookup table makes decoding pure pattern-matching: see the letter, copy the digit. No arithmetic needed.
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.
💡 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.
2.NBT.A.3 Build the lookup table. Write the $10$ letters of $\text{BEST OF LUCK}$ in a row 2.NBT.A.3 Look up each letter of $\text{CLUE}$ in the table. Find $\text{C}$, $\text{L}$, 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.3Read 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.2Read 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.
⭐ 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.
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