A traveller has a gold chain of 7 links joined in a straight line and must pay an innkeeper one link a day for 7 days, settling up each morning with no debt owing and nothing paid in advance. Links may be handed back as change. What is the smallest number of links he needs to cut open?

Cut open the third link only. That single cut leaves three parts: the loose link itself, a piece of 2 links and a piece of 4 links. Day 1: hand over the single link. Day 2: hand over the 2-piece and take the single back as change. Day 3: hand over the single again. Day 4: hand over the 4-piece and take back the single and the 2-piece. Days 5, 6 and 7 then repeat the first three moves. The traps: 2 does work — cutting two links gives pieces of 1, 1, 1, 2 and 2 — but it is one cut more than needed, and the question asks for the fewest; 3 comes from cutting enough links to hand over a fresh single link on several days, which forgets that change can be taken back; 6 cuts every link but one, the plan of someone who never takes change at all.