maths help

Question

The spider web consists of four straight threads each of length 2 metres.
They intersect to form a square of side length 1 metre, with 0.5 metres protruding at either end.
What is the largest number of spiders which can amicably share this spider web, if a spider will
tolerate a neighbour only at a distance of 1.1 metre or more, traveling along the web?

Answer 1

Hi Shivam,

Assume each spider is free to travel everywhere on its cell (a 1mx1m square with two 0.5m protrusions on each of its four sides. Imagine drawing a layer of such 1mx1m squares around the central one (built and occupied by the Spider #1).

There are 8 of them (3x3-1) in that layer.

Each cell would be surrounded by 0.5mx1m strips on each side from two neighboring boxes and thus could be traveled only along that side + two protrusions from its ends.

That layer of boxes clearly cannot be occupied by nrigjbor spiders.

Now, imagine drawing a next two layers of 1m by 1m cells on top of the first layer. From the very drawing of them one could see that there would be only 8 closest cells that the central one could tolerate.

Namely, let's imagine a sort of a "chess board," with the numbers along the horizontal running from -3 to +3 and same ,(-3 to +3) along the vertical. The cell which is two units to the right and three raws below the central cell would have coordinates (+2, -3); the central one would be (0,0).

Then the four CLOSEST cells to the one in the center that would satisfy the minimum 1.1m separation from all points accessible for the central spider would be the ones with coordinates: (0,-4); (4,0); (0,4);(0,-4)

Hope it helps

Best

Dr.Ariel B.