Two coins of each kind (1) P62113


Statement
 

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Given a natural number x and n different coin values c1cn, compute in how many ways it is possible to achieve change x by using each value at most twice. Here, two coins with the same value are considered different.

For example, if x = 4 and the available values are 1 and 2, then there are three ways to achieve it: 1 + 1′ + 2, 1 + 1′ + 2′, and also 2 + 2′.

Input

Input consists of several cases. Every case begins with x and n, followed by c1cn. Assume 1 ≤ n ≤ 20, 1 ≤ cix ≤ 1000, and that all ci are different.

Output

For every case print, in lexicographic order, all possible ways to exactly achieve change x by using each value at most twice. Print every solution with its values sorted from small to big. In doing that, assume 1 < 1′ < 2 < 2′ < …. Use “1p” to print 1′, etcetera. Print a line with 10 dashes at the end of every case.

Hint

A simply pruned backtracking should be enough.

Public test cases
  • Input

    4 2  1 2
    400 1  200
    400 1  300
    5 3  4 2 1
    5 5  1 2 3 4 5
    

    Output

    4 = 1 + 1p + 2
    4 = 1 + 1p + 2p
    4 = 2 + 2p
    ----------
    400 = 200 + 200p
    ----------
    ----------
    5 = 1 + 2 + 2p
    5 = 1 + 4
    5 = 1 + 4p
    5 = 1p + 2 + 2p
    5 = 1p + 4
    5 = 1p + 4p
    ----------
    5 = 1 + 1p + 3
    5 = 1 + 1p + 3p
    5 = 1 + 2 + 2p
    5 = 1 + 4
    5 = 1 + 4p
    5 = 1p + 2 + 2p
    5 = 1p + 4
    5 = 1p + 4p
    5 = 2 + 3
    5 = 2 + 3p
    5 = 2p + 3
    5 = 2p + 3p
    5 = 5
    5 = 5p
    ----------
    
  • Information
    Author
    Salvador Roura
    Language
    English
    Translator
    Albert Atserias
    Original language
    Catalan
    Other languages
    Catalan
    Official solutions
    C++
    User solutions
    C++