Two cubes
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To solve this problem, it is helpful to visualize that a cube has 6 faces and further removal of cubes must grant extra faces. If Kim removes one small cube from a corner of the large cube, the total faces will increase by 3 (since three new faces appear where the cube was removed).
To have the largest cube with 12 faces, Kim must have removed 2 corners. However, this would add 6 faces (3 faces per corner cube removed), resulting in 12 faces total.
Therefore, the smallest number of small cubes that Kim could have removed to create an object with 12 faces is two.
