# Sunday Times Teaser 2823 – Queuing

### by John Owen

#### Published: 30 October 2016 (link)

Tickets for the event went on sale at 0930. The queue started at 0900 when 2 people arrived, then 4 more at 0901, 6 more at 0902 and so on until 60 more arrived at 0929. Just 16 people arrived at 0930, 16 more at 0931, 16 more at 0932 and so on. Tickets were sold steadily at a rate of 25 per minute (one for each person in the queue).

Joe and I were the first to arrive at our respective minutes, but we had identical waiting times before being sold our tickets, and no-one waited for less time to get their ticket. Joe was lucky to get the last ticket to be sold.

At what time did Joe join the queue?

NOTE: This is a clarified version of the text published in the Sunday Times

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Here is a manual solution. Let $$t$$ be the number of minutes after 09:00. Then, for people joining before 09:30, the number of people in the queue ahead of the first person to join at time $$t$$ is $$t(t+1)$$. And this person has a wait time of $30-t+t(t+1)/25 = ((t-12)^2+606)/25$ Hence the minimum wait is 606/25 minutes. For those joining after 09:30, the wait time is $(930-9(t-30))/25 = (1200-9t)/25$Equating this with the earlier minimum wait gives his joining time as 66 minutes, which means that Joe joined the queue at 10:06.