#!/usr/bin/env python3
#
# https://math.stackexchange.com/questions/4271672/given-numbers-1-to-3n-construct-n-equations-of-the-form-a-b-c-or-a


import sys

def debug(*msgs):
    # print(*msgs, file=sys.stderr)
    pass

class SearchState():
    def __init__(this, used, missed, pool):
        this.used = used        # equations selected so far
        this.missed = missed    # set of numbers not yet covered
        this.pool = pool        # unused equations

    def __str__(this):
        return f"{this.used} missing {this.missed}"

    def extend(this):

        # Used a solution
        if len(this.missed) == 0:
            debug("*", this)
            yield this.used
            return

        debug("- extending", this)
        # for each  possible equation that can be used
        # add it to the state and attempt to extend to a complete solution
        pool = set(this.pool)
        for eqn in pool:
            debug(".", eqn)
            if set(eqn) <= this.missed:
                yield from this.add_equation(eqn).extend()
            this.pool.remove(eqn)

    def add_equation(this, eqn):
        # Just like this, but with eqn added to the list of used equations
        # and its elements removed from the "missed" set
        return SearchState(this.used + [eqn],
                           this.missed - set(eqn),
                           this.pool - { eqn })

# equations with numbers up to n
def equations(n):
    eqns = set()
    for i in range(1, n+1):
        for j in range(i+1, n-i+1):
            debug("=", i, j)
            eqns.add((i, j, i+j))
            if i > 1 and i*j <= n:
                eqns.add((i, j, i*j))
    return eqns

if __name__ == '__main__':
    if len(sys.argv) != 2 and len(sys.argv) != 3:
        print("Usage: x3c number [how-many-solutions]", file=sys.stderr)
        exit(2)

    n = int(sys.argv[1])
    z = 12 if len(sys.argv) < 3 else int(sys.argv[2])
    init = SearchState([],      # start with no equations selected
                       set(range(1, n+1)), # numbers 1..n not used yet
                       equations(n)        # equations we can use
                       )
    for result in init.extend():
        print(result)
        z -= 1
        if z <= 0:
            break