Fischer Zoo
Lisa has 8 animal ornaments she wants to arrange on her shelf which has just 8 spots in a row. There are 2 mice, 2 dogs, 2 frogs, 1 giraffe, and 1 elephant. Lisa knows that elephants are afraid of mice, so she wants to arrange the ornaments so that one mouse is somewhere to the left of the elephant and the other mouse is somewhere to the right of the elephant, to trap the elephant between the mice. She also knows that frogs like to jump around, and hence wants the frogs to have an even number of spots between them. How many different ways can Lisa arrange the animals?
Details and assumptions:
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The ornaments of the same type are indistinguishable.
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0 is an even number. So the frogs having 0 spaces between them, i.e. the frogs being adjacent, is also allowed.
The answer is 960.
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1 solution
There are 8 places that the first frog could go. Since there have to be an even number of places between them, there are 4 places for the second frog. We could have also placed these in the other order, so in total there are 16 ways the frogs could be arranged.
Next, there are 6 places that the giraffe could be placed. There are ( 2 5 ) = 1 0 places the dogs could be placed. This leaves 3 spots remaining, and so the elephant must be in the middle and the mice in the other 2. Thus, there are 1 6 × 6 × 1 0 = 9 6 0 different ways the animals could be arranged.
Note: This is actually how many ways there are to arrange the back line in Bobby Fischer's chess variant Fischer Random Chess. The elephant is the king, the giraffe is the queen, the mice are the rooks, the frogs are the bishops, and the dogs are the knights. The bishops must be on opposite colours, so there is an even number of places between them. The king must be between the rooks so that castling is possible in both directions. The reason that this is not just asked about the chess pieces is because Fischer Random Chess is now known as Chess960, named after the number of possible starting positions.