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Use dynamic programming to compute the minimum number of perfect squares for all values up to n, then backtrack to reconstruct the sequence. Alternatively, apply BFS on the graph of remainders to find the shortest path, which naturally yields the sequence. Emphasize that the problem is a variant of the classic coin change problem with squares as coins.
Pro tip: Mention that while the mathematical Lagrange's four-square theorem guarantees the answer is at most 4, the actual sequence still requires computation. Also, note that BFS is often more efficient for finding the sequence because it stops as soon as the target is reached, unlike DP which computes all states.
Confirm that n is a positive integer, and that we need the actual squares, not just the count. Ask about the expected input size to choose the appropriate algorithm.
Decide between dynamic programming and BFS. DP is straightforward for counting and can be adapted for sequence reconstruction; BFS finds the shortest path and naturally reconstructs the sequence.
For DP: initialize an array dp of size n+1 with infinity, set dp[0]=0, and for each i from 1 to n, iterate over squares j^2 <= i, updating dp[i] = min(dp[i], dp[i - j^2] + 1). For BFS: use a queue starting from n, subtract squares until reaching 0, tracking the path.
For DP: after filling dp, start from n and repeatedly find a square j^2 such that dp[n] = dp[n - j^2] + 1, appending j^2 and updating n. For BFS: during traversal, store the parent and the square used to reach each state, then backtrack from 0 to n.
Discuss time and space complexity: DP is O(n√n) time and O(n) space; BFS is O(n) time and space in the worst case. Mention possible optimizations like using a precomputed list of squares or pruning.
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