Problem 22 · 1993 AJHSME
Stretch
Counting & Probability
digit-counting
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?
Show answer
Answer: D — 119.
Show hints
Hint 1 of 2
Every other digit is unlimited — only the 2's can run out. So ignore all other digits and just count how many 2's the page numbers eat as you climb.
Still stuck? Show hint 2 →
Hint 2 of 2
Count 2's in chunks. How many 2's appear writing pages 1–99? (Count the units-place 2's and the tens-place 2's separately.) Then continue past 99 page-by-page until the 22nd two is spent.
Show solution
Approach: count only the 2's, by place value
- The 2's are the bottleneck, so tally them. In pages 1–99, a 2 lands in the units place ten times (2, 12, 22, …, 92) and in the tens place ten times (20–29): 20 twos used, leaving just 2 of them.
- Past 99, the next pages needing a 2 are 102 and 112 — one 2 each. That spends the last two 2's by page 112. Pages 113–119 contain no 2, so they're free, but page 120 would demand a tens-place 2 (a 23rd) that Pat doesn't have.
- So he can number all the way to 119.
- Why this transfers: when one resource is scarce and the rest are free, track only the scarce one and walk forward until it's exhausted — the answer is the last page before the one that would overspend, not the page that breaks the budget.
Mark:
· log in to save