Jigsaw Ratio
In a jigsaw puzzle, pieces are connected horizontally and vertically by a connector--an "out" fitting in an "in". For each connection there are two possible choices: the "out" connector can point left or right, up or down. Assume that this choice is made entirely at random for each connector.
In the puzzle box we find various shapes of pieces. Here are some examples:
(Note: the orientation of a shape is not relevant. For instance, the second drawing represents
all
pieces with three "in" and one "out".)
What is the expected ratio, , of the number of pieces of each shape?
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1 solution
Each interior piece of the puzzle can be described as a tuple of four "ins" and "outs".
There are 16 possible tuples of this kind, and they are all equally likely. So we simply count how many of these tuples match the descriptions given.
p : in-in-in-in
q : in-in-in-out, in-in-out-in, in-out-in-in, out-in-in-in
r : in-in-out-out, in-out-out-in, out-out-in-in, out-in-in-out
s : in-out-in-out, out-in-out-in
It follows immediately that p : q : r : s = 1 : 4 : 4 : 2 .