AMC 10 · 2014 · #24

Grade 6 arithmetic
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Periodic Placement Formula
pattern-recognitionsequences-arithmetictriangular-numbers pattern-recognitionconvert-to-algebraguess-and-check ↑ Prerequisites: sequences-arithmetic
📏 Medium solution 💡 3 insights
Problem
A list of counting numbers is built in rounds. In round n you write down the next n+3 numbers, then jump over (skip) the next n numbers. So round 1 writes 4 numbers and skips 1, round 2 writes 5 and skips 2, and so on, giving 1,2,3,4, 6,7,8,9,10, 13,… Find the 500,000th number that actually gets written down.

Pick an answer.

(A)
$996,\!506$
(B)
$996,\!507$
(C)
$996,\!508$
(D)
$996,\!509$
(E)
$996,\!510$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Tool #5 (Look for a Pattern) turns the wordy rule into a clean block structure: round n contributes n+3 written numbers and n skipped numbers. Tool #4 (Introduce a Variable) names the round number k and builds a formula for how many numbers have been written after k rounds, so we do not have to list a million terms. Tool #6 (Guess and Check) plugs k=996 and k=997 into that formula to pin down which round holds the 500,000th written number. Finally Tool #16 (Change Focus) flips the question to the skipped numbers: the value of a written number equals its position plus how many numbers were skipped before it, and that skipped total is a short triangular-number computation.

1STEP 1

Read the round pattern

Round n writes n+3 numbers then skips n, so blocks run 4,5,6,… gaps run 1,2,3,… and each round eats 2n+3 numbers.

round n: write (n+3), skip n → uses 2n+3 numbers
2STEP 2

Count written numbers after k rounds

After k rounds the written count is 4+5+…+(k+3), a triangular sum: L(k)=(k²+7k)/2.

L(k)=Σ_n=1^k(n+3)=(k(k+1))/2+3k=(k²+7k)/2
3STEP 3

Find the round holding the 500,000th number

(k²+7k)/2 ≈ 500,000 gives k ≈ 996.5, and L(996)=499,494 ＜ 500,000 ≤ 500,494=L(997) — so the target sits in round 997.

L(996)=499,494, L(997)=500,494, 499,494 ＜ 500,000 ≤ 500,494
4STEP 4

Switch focus to the skipped numbers

Flip it: value = position + skips before it. Round 997 skips only after writing, so S=1+2+…+996=(996·997)/2=496,506.

S=Σ_n=1⁹⁹⁶n=(996·997)/2=496,506
5STEP 5

Combine position and skips

Add the skips back: 500,000+496,506=996,506, choice (A) — and it is the only choice ending in 6, as the skip total demands.

500,000+496,506=996,506 → (A)
Answer
996, 506
Cross-check by tracking the natural numbers directly. After round 996, the numbers used up (written plus skipped) total L(996)+S=499,494+496,506=996,000, so round 997 starts writing at 996,001. Round 997 writes 1000 numbers, taking written-positions 499,495 through 500,494. The 500,000th written number is the (500,000-499,494)=506th of these, with value 996,001+505=996,506. Both routes agree, and the answer sits right in the middle of the choice cluster, exactly where a value near 996,500 should land. So (A) is confirmed.
💡Key takeaway

A written number's value is where you are in the counting numbers: add up everything you skipped and stick it onto your position.

  • Read the round pattern
  • Count written numbers after k rounds
  • Find the round holding the 500,000th number
  • Switch focus to the skipped numbers
  • Combine position and skips

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