Circular lawn student trespass problem how many porters needed to catch

Students and porters are banned from going on the lawns.

What is the minimum number of porters required to catch a student who has been on a circular lawn

One, if the porter is good with a lasso.

Bigger than one, maximum speed of porters and students are the same on any surface

Thirty-one porters. Or infinity. Whichever your budget can best afford.

Abstractly-speaking, the student always escapes because the porters are confined to boundary arcs while the student can use straight interior paths. The key principle is that arc lengths exceed chord length. With finitely many porters there are boundary gaps, and once the student gets sufficiently close to the boundary, the remaining straight-line distance to some exit is smaller than the arc distance any porter must cover. You can try this with your fingers or a napkin drawing. Countably many porters fail for a related reason: no countable collection of boundary-confined movers can effectively guard every possible exit point. Blocking all exits requires guarding every boundary point, which is uncountable.

However, in reality, students have shoulders and porters have arms, so nobody guards mere points but chunks of lawn edge. Let's say each porter covers about "2r of boundary" (where r is the reach of one arm), the number of porters spaced around is "N," the student needs a gap at least "w wide" due to the length between their shoulders, and the radius of the lawn is "R." The uncovered gap is roughly 2πR/N−2r. This is our key expression because once that value is smaller than "w," the student is caught by our porters' reach, despite geometry's protest. So, a circular lawn of, let's say as a concrete example, 20 meters in diameter, will need on the order of 25–35 porters to account for various potential sizes of porters and the student. For completion-sake, using global anthropometric ranges and hedging fairly pessimistically, so w=0.41m and r=0.825m (see online anthropometry tables), the largest uncovered gap falls below w at about 31 porters (30.5, rounded up). Thus, you are likely to catch that student on a 20-meter-diameter circular lawn with 31 porters, substantially fewer than an infinite number of porters. Once people have arms and shoulders, infinity is reduced to hyperbole.

Less, porters and students are mathematical points (can be seen but have no size)

Let me guess that on square lawn there need to be 4 porters going for student in coordinate system. For circle we need Pi porters - round up to 4 and this is the number.

Now I think that if porters need to catch student on border of lawn, then it is infinity. If they can catch him anywhere, porters create triangle formation analogical to square coordinates and 3 is enough in this system.

Yes 3 is the flat earth answer (2 if the earth is a sphere)