class Production:
    def __init__(self, prod):
        self.prod = prod
        self.sumSet = set()

    def addSum(self, summation):
        self.sumSet.add(summation)

    def removeSum(self, summation):
        self.sumSet.remove(summation)

    def __str__(self):
        return "Prod object: {0} {1}".format( self.prod, self.sumSet)

class Summation:
    def __init__(self, prod):
        self.sum = prod
        self.prodSet = set()

    def addProd(self, summation):
        self.prodSet.add(summation)

    def removeSum(self, summation):
        self.prodSet.remove(summation)

    def __str__(self):
        return "Sum object: {0} {1}".format( self.sum, self.prodSet)

class Graph:
    def __init__(self, minNum, maxNum):
        self.sumDict = dict()
        self.prodDict = dict()
        for a in range(minNum, maxNum+1):
            for b in range(a, maxNum+1):
                self.sumDict.setdefault(a+b, Summation(a+b))
                self.prodDict.setdefault(a*b, Production(a*b))
                self.sumDict.get(a+b).addProd(a*b)
                self.prodDict.get(a*b).addSum(a+b)

    def allProdsWithMultipleSums(self, summ):
        prodSet = self.sumDict[summ].prodSet
        for prod in prodSet:
            if len(self.prodDict[prod].sumSet) <= 1:
                return False
        return True

    def removeSum(self, summ):
        prodSet = self.sumDict[summ].prodSet
        self.sumDict.pop(summ, None)
        for prod in prodSet:
            self.prodDict[prod].sumSet.remove(summ)
            if len(self.prodDict[prod].sumSet) == 0:
                self.prodDict.pop(prod, None)

    def removeProd(self, prod):
        sumSet = self.prodDict[prod].sumSet
        self.prodDict.pop(prod, None)
        for summ in sumSet:
            self.sumDict[summ].prodSet.remove(prod)
            if len(self.sumDict[summ].prodSet) == 0:
                self.sumDict.pop(summ, None)

graph = Graph(minNum=2, maxNum=99)

sumsToRemove = {summ for summ in graph.sumDict.keys() if not graph.allProdsWithMultipleSums(summ) }
for summ in sumsToRemove:
    graph.removeSum(summ)

prodsToRemove = {prod for prod in graph.prodDict.keys() if len(graph.prodDict.get(prod).sumSet) != 1}
for prod in prodsToRemove:
    graph.removeProd(prod)

for s in graph.sumDict.values():
    if len(s.prodSet) == 1:
        print("sum: {0}".format(s.sum))
        print("prod: {0}".format(s.prodSet.pop()))