The Bus Timetable Puzzle
A regional town has two bus routes that share a stop.
- Route A buses arrive every 12 minutes, starting at 7:00 am.
- Route B buses arrive every 18 minutes, starting at 7:00 am.
- Both routes stop running at 7:00 pm (the last bus of each route leaves at exactly 7:00 pm).
Mia arrives at the stop at 8:05 am. She will take whichever bus comes first. A trip on Route A takes 25 minutes; a trip on Route B takes 40 minutes. Mia needs to reach the library, which is a 5-minute walk from the stop at the end of either route.
Her friend says: 'Just wait for Route A-it's always faster.'
1. At what times after 8:05 am do the next Route A and Route B buses arrive?
Read the worked answer
Route A buses are at 7:00, 7:12, 7:24, … i.e. times that are multiples of 12 minutes after 7:00.
From 7:00 to 8:05 is 65 minutes. 65 ÷ 12 = 5 remainder 5, so the last A bus was at 8:00 and the next is at 8:12 (7 minutes' wait).
Route B buses are multiples of 18 minutes after 7:00. 65 ÷ 18 = 3 remainder 11, so the last B bus was at 7:54 and the next is at 8:12 (7 minutes' wait).
So both buses arrive at 8:12 am-they coincide. This is because 12 and 18 share a common multiple of 36, and 7:00 + 72 minutes = 8:12 is a shared arrival.
2. Which bus should Mia take, and does the friend's advice hold?
Read the worked answer
Both arrive at 8:12, so Mia boards whichever she prefers-but the arrival differs.
- Route A: 8:12 + 25 min ride + 5 min walk = 8:42 am.
- Route B: 8:12 + 40 min ride + 5 min walk = 8:57 am.
Route A gets her there 15 minutes earlier, so on this occasion the friend's advice happens to be right.
But the friend's claim is 'always faster', which is a stronger statement. It is only true when the wait times are equal or Route A's wait is not much longer. If Mia had arrived at 8:05 and the A bus had just left (say she arrived at 8:01), she might wait 11 minutes for A versus 0 for B, and B's longer ride could still win. The judgement is to separate 'true today' from 'always true'-the friend has confused a lucky coincidence with a rule.