Blame for 948.md

## Page 1

948
Fibonaccimal Base
The well known Fibonacci sequence is obtained by starting with 0 and 1 and then adding the two last
numbers to get the next one. For example the third number in the sequence is 1 (1=1+0), the forth is
2 (2=1+1), the fifth is 3 (3=2+1) and so on.
i
0
1
2
3
4
5
6
7
8
9
Fib(i)
0
1
1
2
3
5
8
13
21
34
Figure 1 - The first numbers in the Fibonacci sequence
The sequence appears on many things in our life, in nature, and has a great significance. Among
other things, do you know that all positive integer numbers can be represented as a sum of numbers in
the Fibonacci sequence? More than that, all positive integers can be represented as a sum of a set of
Fibonacci numbers, that is, numbers from the sequence, without repetition. For example: 13 can be the
sum of the sets {13}, {5,8} or {2,3,8} and 17 is represented by {1,3,13} or {1,3,5,8}. Since all numbers
have this property (do you want to try to prove this for yourself?) this set could be a nice way to use as
a "base" to represent the number. But, as we have seen, some numbers have more than one set whose
sum is the number. How can we solve that? Simple! If we add the constraint that the sets cannot
have two consecutive Fibonacci numbers, than we have a unique representation for each number! This
restriction is because the sum of any two consecutive Fibonacci numbers is just the following Fibonacci
number.
Now that we know all this we can prepare a nice way to represent any positive integer. We will use
a binary sequence (just zeros and ones) to do that. For example, 17 = 1 + 3 + 13 (remember that no
two consecutive Fibonacci numbers can be used). Let’s write a zero for each Fibonacci number that is
not used and one for each one that is used, starting at the right. Then, 17 = 100101. See figure 2 for a
detailed explanation. In this representation we should not have zeros at the left, this is, we should only
write starting with the first one. In order for you to understand better, note that in this scheme, not
using two consecutive Fibonacci numbers means that the binary sequence will not have two consecutive
ones. When we use this representation for a number we say that we are using the Fibonaccimal base,
and we write it like 17 = 100101 (fib).
17 =
1
0
0
1
0
1
13+3+1 =
13
8
5
3
2
1
Figure 2 - Explaining the representation of 17 in Fibonaccimal base
Given a set of numbers in decimal base, your task is to write them in the Fibonaccimal base.
Input
The first line of input contains a single number N, representing the quantity of numbers that follow
(1 ≤N ≤500).
Than follow exactly N lines, each one containing a single positive integer smaller than 100 000 000.
These numbers can come in any order.
Output
You should output a single line for each of the N integers in the input, with the format ‘DEC BASE
= FIB BASE (fib)’. DEC BASE is the original number in decimal base and FIB BASE is its
representation in Fibonaccimal base. See the sample output for an example.


## Page 2

Sample Input
10
1
2
3
4
5
6
7
8
9
10
Sample Output
1 = 1 (fib)
2 = 10 (fib)
3 = 100 (fib)
4 = 101 (fib)
5 = 1000 (fib)
6 = 1001 (fib)
7 = 1010 (fib)
8 = 10000 (fib)
9 = 10001 (fib)
10 = 10010 (fib)