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Chemists work with periodic table elements, but here at Code Jam, we have been using our advanced number smasher to study googlements. A googlement is a substance that can be represented by a string of at most nine digits. A googlement of length L must contain only decimal digits in the range 0 through L, inclusive, and it must contain at least one digit greater than 0. Leading zeroes are allowed. For example,
001 are valid googlements of length 3.
400 (which contains a digit, 4, greater than the length of the googlement, 3) and
000 (which contains no digit greater than 0) are not.
Any valid googlement can appear in the world at any time, but it will eventually decay into another googlement in a deterministic way, as follows. For a googlement of length L, count the number of
1s in the googlement (which could be 0) and write down that value, then count the number of
2s in the googlement (which could be 0) and write down that value to the right of the previous value, and so on, until you finally count and write down the number of Ls. The new string generated in this way represents the new googlement, and it will also have length L. It is even possible for a googlement to decay into itself!
For example, suppose that the googlement
0414 has just appeared. This has one
3s, and two
4s, so it will decay into the googlement
1002. This has one
3s, and zero
4s, so it will decay into
1100, which will decay into
2000, which will decay into
0100, which will decay into
1000, which will continuously decay into itself.
You have just observed a googlement G. This googlement might have just appeared in the world, or it might be the result of one or more decay steps. What is the total number of possible googlements it could have been when it originally appeared in the world?
The first line of the input gives the number of test cases, T. T test cases follow. Each consists of one line with a string G, representing a googlement.
For each test case, output one line containing
Case #x: y, where
x is the test case number (starting from 1) and
y is the number of different googlements that the observed googlement could have been when it first appeared in the world.
3 20 1 123
Case #1: 4 Case #2: 1 Case #3: 1
In sample case #1, the googlement could have originally been
20, or it could have decayed from
11, which could have itself decayed from
21. Neither of the latter two could have been a product of decay. So there are four possibilities in total.
In sample case #2, the googlement must have originally been
1, which is the only possible googlement of length 1.
In sample case #3, the googlement must have been
123; no other googlement could have decayed into it.