A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, one single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying n cuts is denoted by x_n. The experiment is performed on a solid cube where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is **not** a possible value for x_4?<br/>

Possible cases:

• x4=5: make 4 identical slabs with 3 equally spaced parallel cuts; 4th cut splits only one slab.
• x4=8: same 4 slabs; 4th cut passes through all 4.
• x4=12: make 6 identical cuboids using cuts x=1/3, x=2/3 and y=1/2; 4th cut z=1/2 splits all 6.

For x4=16, after 3 cuts we must already have 8 pieces, and the 4th cut must split all 8. Now look inside the 4th cutting plane: the first 3 cuts appear there as at most 3 lines. Three lines can divide a plane into at most 7 regions, so the 4th plane can intersect at most 7 existing pieces. Hence 16 cannot occur.