Some thoughts

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    This is a 4 partition problem. An easier 2 partition problem is Partition Equal Subset Sum. Thanks to @withacup. Following is the nicest I have

        bool makesquare(vector<int>& nums) {
            int tot = accumulate(nums.begin(),nums.end(),0);
            if(!tot || tot%4) return 0;
            int edge[4]={0};
            sort(nums.begin(),nums.end(),greater<int>()); //try numbers from big to small
            return isSqr(0, tot/4, edge,nums);
        bool isSqr(int p, int sum, int edge[4], vector<int>& nums) {
            if(p==nums.size()) return edge[0]==sum && edge[1]==sum && edge[2]==sum;
            unordered_set<int> ht;
            for(int i=0;i<4;i++) {
                const auto &pr = ht.insert(edge[i]); //do not create duplicate states from a state
                if(!pr.second || nums[p]+edge[i]>sum) continue; //do not expand to an invalid state
                if(isSqr(p+1,sum,edge,nums)) return 1;
            return 0;

    There are 3 optimizations based on brute force.

    1. Do not expand to an invalid state with a larger than average partition.
    2. Try the numbers from big to small. This is not as easy as it seems. During DFS, we have 4 bins and we try to put a number in each of them to generate the next state. Worst case is O(4^n) which happens when a state always expands to 4 valid state, that is when expand from small numbers to larger numbers. In this case, we generate a lot of states from the small numbers and figure out they are useless because they cannot accommodate the large numbers. If we search from large numbers, then we may have a filled bin at first try and runtime becomes O(3^n).
    3. Do not create duplicate states from a state. A state is determined by the 4 bins and the current number. If memory allows, we can have a 5d memoization array to remove redundancy. Since a state is defined by 5 variables, it may not be very often to generate a duplicate state so it is probably ok to go without memorization and use the cheap trick.

    What if the numbers are arbitrary? Looks like only #3 still works...

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