Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 127

[михаил потапыч]

GaryKa:
Although no, neither straight nor spiral is suitable, the forest can be spiral or straight. Here you should probably use some kind of self-intersecting curve to cut off areas reliably, i.e. use the fact that the forest is solid, without glades.

A forest can do anything. And in the abstract the right answer will still be as a variant of the (worst) passage of the whole forest.

[GaryKa]

Mathemat:

(4) Megamogg is in a dense forest that covers exactly 100 km2 in area. The shape of the forest is unknown, but the forest is solid, without glades. Megamogg wants to get out of the forest by walking the minimum possible distance. What minimum length (and shape) path guarantees that it will be able to find the border of the forest?

I'll try to give the answer, but it is suspiciously simple.)

A circle is a flat figure which has the following property: the perimeter of the given figure (circle) is minimal among all the figures with the given area. If we move around the circle, then by traversing the whole circle we will cut off (bypass) the area by the minimal trajectory. The area of the forest is 100 km, then Megabrain must move on a circle with radius = 10/sqrt(Pi). Hence, a path (circle) with minimum length = 20*sqrt(Pi) guarantees that it will be able to find the border of the forest.

[Igor Maslov]

The option is to walk in a circle with a radius of 5km. At the most, you will walk almost the entire circle (a square with a cut at the edge), i.e. approximately 31.4 km.

lied, corrected

[михаил потапыч]

muallch:

The option is to walk in a circle with a radius of 5km. At the most, you will walk almost the entire circle (a square with a cut at the edge), i.e. approximately 31.4 km.

lied, corrected

Exactly, but it's not 5.65.

[Igor Maslov]

Mischek:
Right, but it's not 5.65.

Why? If the forest is a square with a small (hardly affecting the total area) slot in the middle of one side, and MM stands in the corner of this slot, then walking on the circle inscribed in the square will come to the other side of the slot and will exit into it. The radius of the inscribed circle is 5km.

[михаил потапыч]

muallch:
Why? If the forest is a square with a small (having almost no effect on the total area) slot in the middle of one side, and MM stands in the corner of this slot, then walking on the circle inscribed in the square will come to the other side of the slot and exit into it. The radius of the inscribed circle is 5km.

And if the circular

[Igor Maslov]

Mischek:
And if it's round...

Ё... Exactly!

[михаил потапыч]

Here 's the challenge, but in the form of a finished game

[Sceptic Philozoff]

Mischek: Here 's the challenge, only in the form of a finished game.

Yeah, it's interesting. You have to go in from afar. But it doesn't always seem to be solved either.

[Sergey Gridnev]

It always seems to be solved by getting the circles right from the start.