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This work proposes a new method to sequentially train deep neural networks on multiple tasks without suffering catastrophic forgetting, while endowing it with the capability to quickly adapt to unseen tasks. Starting from existing work on network masking (Wortsman et al., 2020), we show that simply learning a linear combination of a small number of task-specific supermasks (impressions) on a randomly initialized backbone network is sufficient to both retain accuracy on previously learned tasks, as well as achieve high accuracy on unseen tasks. In contrast to previous methods, we do not require to generate dedicated masks or contexts for each new task, instead leveraging transfer learning to keep per-task parameter overhead small. Our work illustrates the power of linearly combining individual impressions, each of which fares poorly in isolation, to achieve performance comparable to a dedicated mask. Moreover, even repeated impressions from the same task (homogeneous masks), when combined, can approach the performance of heterogeneous combinations if sufficiently many impressions are used. Our approach scales more efficiently than existing methods, often requiring orders of magnitude fewer parameters and can function without modification even when task identity is missing. In addition, in the setting where task labels are not given at inference, our algorithm gives an often favorable alternative to the one-shot procedure used by Wortsman et al., 2020. We evaluate our method on a number of well-known image classification datasets and network architectures.