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Rotation is one of several popular pocket billiards games. It uses 15 balls numbered from 1 to 15, and set them up as illustrated in the following figure at the beginning of a game. (Note: the ball order is modified from real-world Rotation rules for simplicity of the problem.)
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You are an engineer developing an automatic billiards machine. For the first step you had to build a machine that sets up the initial condition. This project went well, and finally made up a machine that could arrange the balls in the triangular shape. However, unfortunately, it could not place the balls in the correct order.
So now you are trying to build another machine that fixes the order of balls by swapping them. To cut off the cost, it is only allowed to swap the ball #1 with its neighboring balls that are not in the same row. For example, in the case below, only the following pairs can be swapped: (1,2), (1,3), (1,8), and (1,9).
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Write a program that calculates the minimum number of swaps required.
The first line of each test case has an integer $N$ ($1 \leq N \leq 5$), which is the number of the rows.
The following $N$ lines describe how the balls are arranged by the first machine; the $i$-th of them consists of exactly $i$ integers, which are the ball numbers.
The input terminates when $N$ = 0. Your program must not output for this case.
For each test case, print its case number and the minimum number of swaps.
You can assume that any arrangement can be fixed by not more than 45 swaps.
2 3 2 1 4 9 2 4 8 5 3 7 1 6 10 0
Case 1: 1 Case 2: 13