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We present a strongly fault-tolerant design for the k-ary n-cube multiprocessor and examine its reconfigurability. Our design augments the k-ary n-cube with (k/j)n spare nodes. Each set of jn regular nodes is connected to a spare node and the spare nodes are interconnected as either a (k/j)-ary n-cube if j≠(k/2) or a hypercube of dimension n if j=k/2. Our approach utilizes the capabilities of the wave-switching communication modules of the spare nodes to tolerate a large number of faulty nodes. Both theoretical and experimental results are examined. Compared with other proposed schemes, our approach can tolerate significantly more faulty nodes with a low overhead and no performance degradation.