Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #2
Grade 4 arithmeticPick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 BEST OF LUCK controls the answer.
Build the letter-digit table
Build the lookup table: write BEST OF LUCK in a row with 0, 1, …, 9 underneath, one digit per letter.
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.
Writing the ten letters of BEST OF LUCK in a row and placing the digits 0 through 9 beneath them in the order the letters appear gives every letter exactly one digit.
▸ Why?
The phrase has exactly ten letters and there are exactly ten digits from 0 to 9, so pairing them off in order leaves no letter without a digit and never hands one digit to two different letters.
▸ Why?
Two groups that hold the same number of things can be matched thing for thing, so each item on one side lines up with a single partner on the other and none is left over.
Look up each letter
Look up each letter of CLUE in the table: C is 8, L is 6, U is 7, E is 1.
The lookup table makes decoding pure pattern-matching: see the letter, copy the digit. No arithmetic needed.
2.NBT.A.3Make A Systematic ListWrite the digits in order
Write the digits in CLUE's order — thousands, hundreds, tens, ones — to get 8671.
Reading a 4-digit number is just stacking thousand-hundred-ten-one place values in order — the same structure as reading CLUE left to right.
4.NBT.A.2Change Focus Count The ComplementWhen 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.
- Build the letter-digit table
- Look up each letter
- Write the digits in order
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