It is one of those travel decisions that regularly leaves us stumped. When standing at a bus stop for a short journey, is it better to stick it out and wait or start walking?

The answer, according to mathematicians, will delight couch potatoes everywhere. It almost always pays to wait for the next bus, however enticing the walk might seem.

Only when no bus is due for an hour or more and your destination is less than a kilometre away is it worth walking, the American researchers said. Giving in to your frustration and starting the journey on foot will normally just make you late.

“Many mathematicians probably ponder this on their way to work, but never get round to working it out,” Scott Kominers, of Harvard University, told New Scientist magazine. “It certainly has changed the way I travel.” Dr Kominers joined Justin Chen, of the California Institute of Technology, and Robert Sinnott, also of Harvard, who had considered the issue while working on a problem together.

In their paper, Walk Versus Wait: The Lazy Mathematician Wins, the researchers wrote: “Recently, the first author [Justin] had to walk to the second author’s [Scott’s] house to work on a problemset.

“There is a bus route which covers the distance directly, but the bus arrives sporadically. So, he faced a conundrum: walk the distance or wait for the bus? Being lazy, he would always rather ride the bus, if possible.

Being punctual, however, he will always choose the option which gets him to his destination as quickly as possible.”

The mathematicians then worked out the variables for the journey and turned them into a formula. It takes into account d, the distance in miles Dr Chen has to travel; n, the number of bus stops; vw, Dr Chen’s walking speed; and vb, the bus’s speed. The time when the journey starts – t – equals 0, and time spent waiting is tw.

The formula shows that it is never worth him starting to walk and then boarding the bus at the next stop if it happens to be passing: all he does is waste energy or see a bus speeding past him between stops.

The time spent waiting added to the time the bus would take is almost always lower than the time it would take him to walk. “‘Eureka!’, Justin shouted upon realising that the laziest possible waiting strategy would prevail,” the paper concludes. “He promptly sat down to wait for the bus.”

There are circumstances, however, in which the formula does not quite work. “Being a mathematician, Justin had, of course, chosen to ponder the problem, entirely forgetting the planned meeting,” the researchers wrote. “In the meantime, Scott had grown tired of waiting and had walked to Justin’s house. When he arrived, he pointed out that it was a holiday, hence no buses were running.”

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