AMC 10 · 2021 · #1
Grade 4 arithmeticPick an answer.
Straight column addition works, but the four addends are cyclic shifts of the same four digits, and Tool #5 (Look for a Pattern) turns that observation into the whole solution. Tool #15 (Organize Information in More Ways) is what makes the pattern visible: stack the numbers and read down each place-value column instead of reading across each number. Tool #7 (Identify Subproblems) then splits the total into four independent column sums, each of which is the same easy sum 1 + 2 + 3 + 4.
Stack the numbers and read down
Stack them and read down the columns.
Reading a column of stacked numbers tells you which digits are competing for the same place value.
4.NBT.A.2Organize Information In More WaysEvery column holds 1, 2, 3, 4
Every column holds each digit exactly once.
A cyclic shift moves every digit through every place, so no place gets favored.
A cyclic shift moves every digit through every place, so no place gets favoured.
▸ Why?
After a full round of shifts each digit is back where it began, so each has visited every place once.
▸ Why?
A number is its digits weighted by their places, so equal visits mean equal contributions.
Rebuild the total from the columns
Rebuilding from the columns gives 11,110.
If every place is worth the same count, factor that count out and multiply once instead of adding four times.
4.NBT.B.4Identify SubproblemsWhen numbers are made of the same digits shuffled around, add down the columns instead of across the rows: every column carries the same digit sum, and the total falls out at once.
- Stack the numbers and read down
- Every column holds 1, 2, 3, 4
- Rebuild the total from the columns