Make the still runner lonely

1 · Push two runners away

Beginner

That largest gap is the maximum loneliness ML(V). What does the conjecture predict with n moving runners?

At least 1/(n + 1) of a lap At least 1/(2n) of a lap At least 1/n of a lap

2 · Add a third runner

Intermediate

According to the article, what is the maximum loneliness of speeds 1, 2, 3 and 4?

One eighth of a lap One fifth of a lap One quarter of a lap

3 · Open the danger windows

Intermediate

With n moving runners, how far does the counting argument guarantee a free moment?

As long as δ < 1/n As long as δ < 1/(n + 1) As long as δ < 1/(2n)

4 · Go to a hundred runners

Expert

With n = 100 moving runners, the easy bound guarantees 1/200 of a lap. What does the new theorem guarantee?

More than (1 + c)/200, with the same c as for 3 runners At least 1/101, the value the conjecture predicts A gain of order 1/100², which fades as n grows

5 · Keep a single runner

Expert

Alone, this runner gets at best half a lap away: exactly 1/(2n) for n = 1. Why does the new theorem not contradict this?

The constant c drops to zero when n is 1 The theorem needs at least two runners The conjecture itself fails for n = 1

Runner left alone

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