The chances are that you have come to this page because you are having difficulty solving one of my duck balancing puzzles on http://www.geocaching.com.
If not you may want to try to solve the puzzles first.
Ducks of different species sit on perches that are suspended from the roofs of their duck houses. Each species is a different weight (you can ignore the weight of the perches) and your job is to work out which species fill vacant spaces indicated by question marks.
First a little basic physics. The further away a duck sits from the fulcrum (the hanging point) the more it will tip the perch – so if two ducks balance a perch and one is sitting one position away from the fulcrum whilst the other is three positions away, the one nearest the fulcrum weighs three times more.
You firstly need to determine the relative weights of each species of duck. To do this identify a perch where there are no question marks – in the example below the lower perch – and look at this in isolation.
Firstly you can ignore the yellow duck standing on the fulcrum as, clearly, this does not tip the lower perch in either direction. This leaves one yellow duck one position to the left which is balanced by two blue ducks to the right. Following the principle described above, the right hand blue duck is twice the distance from the fulcrum so effectively counts as two ducks in terms of the leverage it exerts – so in total three blue ducks are the equivalent of one yellow duck (which counts as a single being only one position away).
So a yellow duck weighs three times more than a blue one (1 yellow duck = 3 blue ducks, although I’ll try to avoid algebra).
From now one we will work in the lowest unit – i.e. blue ducks. In the illustration below I have shown in “blue duck units”, B, the relative values of the ducks when considering the effect on the lower perch. The left hand yellow duck is worth 3 blue, the other yellow duck has no effect and therefore no value, the blue duck one position away counts as one blue, and the far right hand duck is worth 2 blues. We now have to look at the effect of the lower perch on the upper one.
The position of the birds along the lower perch has no effect on the upper perch, but the point at which it is attached to the upper perch is significant. Add up the weights (in blue duck units) of the birds on the lower perch and the total comes to 8 as shown in red figures in the box. However, as the perch is attached two positions away on the top perch, the relative weight is 16 blue ducks.
Now look at the right hand side of the top perch remembering that a yellow duck weighs three times a blue one. So the two ducks are worth 6 and 9 blues as shown – a total of 15 blues. To balance we need one more blue duck at the first position so the missing duck is blue!