### Brain Teasers

# Ringing All Around

It's easy to see that a ring can completely hold (surround) two identical smaller rings with half the diameter, without overlapping. Three times the diameter, the bigger ring is space enough to seat seven rings; the outer six touching both the middle ring and the bigger circle/perimeter.

Using this basic information and your imagination, determine the maximum number of rings that could be housed inside another ring with four, five, six and seven times the diameter.

Using this basic information and your imagination, determine the maximum number of rings that could be housed inside another ring with four, five, six and seven times the diameter.

### Hint

I would say "go and chop down your trees" but Greenpeace people might sue me.### Answer

For 4D (four times the diameter) ring, place the 2D ring exactly at the center inside 4D, and you will have donut-shaped empty space with a width of a D ring, around the 2D. The number of rings to be fit in this vacancy, R, is determined from the formulaR < (n - 1)*pi

where n is the nth times the diameter and R is an integer.

This formula is derived from the fact that the circumference of a polygon with R sides (made up by connecting all the nucleus of the rings in the donut area) is always smaller (actually is inscribed in) than the locus of the donut area. This locus is, of course, neither the donut's outer nor inner ring, but the one in between those two.

In a 4D ring:

central 2D = 2

R < (4 - 1)*pi = 9

number of rings, x = 2 + 9 = 11

In 5D ring with 3D center:

x = 7 + 12 = 19

6D: x = 11 + 15 = 26

7D: x = 19 + 18 = 37

The center for nth ring is always the (n - 2)th one. This calculation is based on the hypothesis that the accumulated/shared empty spaces (outside small rings) among central and donut area is small enough to fit for another ring.

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