AMC 8 · 2019 · #13
Grade 4 number-theoryPick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The two-digit palindromes form a tiny pattern: 11, 22, 33, …, 99 — Tool #5 (Pattern) immediately spots that every one of them equals 11 × k for k = 1, …, 9. That single observation forces the sum N to be a multiple of 11, which collapses the search space from 900 three-digit numbers down to a short list. Tool #2 (Systematic List) then walks the three-digit multiples of 11 in order (110, 121, 132, …) to find the smallest one that is NOT a palindrome. Finally Tool #6 (Guess and Check) confirms that the winning candidate 110 can actually be written as a sum of three DISTINCT two-digit palindromes. We deliberately avoid Tool #13 (Algebra) — the pattern + listing path is much cleaner.
List every two-digit palindrome — the two digits must match, so there are only nine of them.
Understanding that a two-digit number is "tens digit, then ones digit" is the Grade 1 place-value idea — it tells us exactly which two-digit numbers can be palindromes.
1.NBT.B.2Make A Systematic ListSpot the pattern: each palindrome aa equals 10a + a = 11a, so all nine are multiples of 11.
Rewriting 10a + a as 11 · a is the Grade 4 multi-digit multiplication / distributive-property move that turns the list into one clean pattern.
4.NBT.B.5Look For A PatternSince N sums three multiples of 11, N itself is a multiple of 11 — so search only three-digit multiples of 11.
Adding multiples of 11 gives another multiple of 11 — a Grade 4 "multiples of a whole number" property that prunes the search.
4.OA.B.4Look For A PatternScan three-digit multiples of 11 upward: 110 is not a palindrome (121 is), so the least candidate is 110.
Reading the hundreds, tens, and ones digits of a three-digit number to check whether it reads the same backwards is exactly the Grade 2 place-value skill.
2.NBT.A.1Make A Systematic ListConfirm 110 is buildable: 11 + 22 + 77 = 110 uses three distinct two-digit palindromes.
Adding three two-digit numbers to hit a target within 1000 is a Grade 2 addition strategy — exactly what guess-and-check needs here.
2.NBT.B.7Guess And CheckAdd the digits of N = 110: 1 + 1 + 0 = 2 → (A).
Adding three single digits is the Grade 1 "sum of three whole numbers within 20" standard.
1.OA.A.2Look For A PatternThis AMC 8 problem only needs Grade 4 "multiples of a whole number" you already know — once you see that every two-digit palindrome is a multiple of 11, the whole puzzle solves itself!