Photo:
Michael Niestedt / DER SPIEGEL
Six gears are arranged to form a concave, irregular hexagon.
Each gear has a diameter of ten centimeters, the distances between the gears are shown in the following sketch.
Icon: enlargePhoto: Michael Niestedt / DER SPIEGEL
You want to put a chain around the six gears.
This should fit tightly, i.e. not sag between the gears.
How long does this chain have to be?
Note on the diameter of ten centimeters: When a chain rests on a gear, the center points of the chain pins are exactly five centimeters from the center of a gear. The bolts are located on the left and right side of a chain link. On a gear, they always sit exactly between the teeth of the gear.
Please scroll down for the solution!
Photo: Michael Niestedt / DER SPIEGEL
solution
The chain has a length of about
284.4 centimeters
.
It is very easy to calculate the length of the straight chain elements.
This is simply the sum of the distances between the six gears - so:
41 + 42 + 41 + 44 + 43 + 42 = 253 cm
But how long are the parts of the chain that rest on the sprockets?
Obviously, their length depends on the angle by which the chain changes its direction on the respective gear.
The problem is that we don't know exactly what these angles are.
But we don't have to, because the six angles add up to exactly 360 degrees - see the following sketch:
Icon: enlargePhoto: Michael Niestedt / DER SPIEGEL
In order to calculate the total length of the chain, we simply add to the straight chain sections (253 cm) the chain length that is necessary for a complete circling of a gear.
This length is pi times the diameter, i.e. 3.14 * 10 cm = 31.4 cm.
The result is 253 + 31.4 = 284.4 cm.
The idea for this puzzle comes from the book "Alice im Knobelland" by Heinrich Hemme.
If you missed a puzzle from the past few weeks, here are the ten most recent episodes:
Total known, numbers sought
The bar castling
Balanced record collection
How many flowers are there?
The clever carpet seller
Lots of lies and a wedding
The coin with the hole
The lost time
The nibbled hexagon
Small circle encircles large circle
Icon: The mirror