Problem 22 · 2012 Math Kangaroo
Stretch
Logic & Word Problems
work-backward
A goldsmith has 12 double-links of chain. Out of these he wants to make a single closed chain with 24 links. What is the minimum number of links that he must open (and close again)?
Show answer
Answer: A — 8
Show hints
Hint 1 of 2
Opening both links of one whole double-link frees two connectors.
Still stuck? Show hint 2 →
Hint 2 of 2
How many pieces must you sacrifice so the freed links join all the rest into one loop?
Show solution
Approach: sacrifice whole pieces to use their links as connectors
- There are 12 double-links (pieces). Opening both links of one piece gives two open links that can join other pieces, while using up that whole piece.
- If you open p whole pieces you get 2p connector links; the remaining 12 − p pieces need 12 − p joins to close into one loop.
- Require 2p ≥ 12 − p, i.e. 3p ≥ 12, so p ≥ 4; opening 4 pieces means opening 2 × 4 = 8 links (A).
Mark:
· log in to save