Problem 5 · 2000 AMC 8
Easy
Logic & Word Problems
work-backwardcareful-counting
Each principal of Lincoln High School serves exactly one 3-year term. What is the maximum number of principals this school could have during an 8-year period?
Show answer
Answer: C — 4.
Show hints
Hint 1 of 2
8 ÷ 3 isn't a whole number, so the terms don't line up neatly with the window. The trick to MAXIMIZING is to let the window catch the tail end of one term and the front end of another — waste partial terms at both edges.
Still stuck? Show hint 2 →
Hint 2 of 2
To maximize how many things overlap a fixed window, push a boundary just inside each end. The two end-principals only need to touch the window by a single year.
Show solution
Approach: let partial terms hang over both ends of the window
- Imagine year 1 is the *final* year of some principal's term — that principal counts, even though most of their term was before our window.
- Their 3-year terms then cover years 2–4 (principal 2) and 5–7 (principal 3). That uses years 2 through 7.
- Year 8 is the *first* year of a fourth principal — they count too. Total: 4 principals.
- The principle: to fit the most fixed-length blocks into a window, align a block boundary just inside each end so the two end blocks only barely overlap — you gain a partial block at each edge. (Choice E's '8' is the trap: a 3-year term can never be just 1 year.)
Mark:
· log in to save