AMC 10 · 2009 · #5

Grade 5 number-theory
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence
pattern-recognitiondigit-sumperfect-squares easier-related-problempattern-recognition ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 2 insights
📘 View easy version →
Problem
The number 111111111 is made of nine 1s. Square it, then add up every digit of the result. What is that digit sum?

Pick an answer.

(A)
$\ 18$
(B)
$\ 27$
(C)
$\ 45$
(D)
$\ 63$
(E)
$\ 81$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Squaring 111111111 directly is heavy, so first shrink the problem (Tool #9): square the small repunits 11, 111, 1111 where the arithmetic is easy. Those small squares reveal a clean shape — a number that counts up then back down — which is exactly what Tool #5 (Look for a Pattern) is for. Once the pattern gives the full square, adding its digits splits into a tidy subproblem (Tool #7): a run of 1 up to 9 and back down, which sums to a perfect square.

1STEP 1

Square the small repunits

Shrink the problem: square 11, 111, and 1111 by hand. Each result reads the same forwards and backwards.

11²=121, 111²=12321, 1111²=1234321
2STEP 2

Spot the count-up-count-down pattern

Line them up: 1, 121, 12321, 1234321. While n ≤ 9, n ones square to digits climbing 1 up to n then mirroring back down.

1, 121, 12321, 1234321, … → climb 1 → n then back to 1
3STEP 3

Apply the pattern to nine 1s

Here n=9, the last case before carries begin, so the square is 12345678987654321.

111111111² = 12345678987654321
4STEP 4

Add the digits

Add its digits: 1+2+…+9+…+2+1 = 36+9+36 = 81, choice (E).

1+2+…+9+…+2+1 = 9² = 81 (E)
Answer
81
The answer 81 is one of the choices, and it equals 9² — the square of the number of 1s — which is a memorable, clean result. A sanity check on scale: the square 12345678987654321 has 17 digits, and an average digit near 5 would predict a sum around 85, right where 81 sits. Smaller cases confirm the rule: 111²=12321 has digit sum 9=3², and 1111²=1234321 has digit sum 16=4², so nine 1s giving 9²=81 fits the family exactly.
💡Key takeaway

Squaring a string of 1s makes the digits count up then back down, so nine 1s square to 12345678987654321 whose digits add to 9²=81.

  • Square the small repunits
  • Spot the count-up-count-down pattern
  • Apply the pattern to nine 1s
  • Add the digits

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