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?

Correct Option: (d) 5

We are cutting a solid cube using n cuts, where pieces remain unmoved after each cut. The maximum number of pieces produced by n cuts of a cube is given by the formula x_n = (n^3 + 5n + 6) / 6. Let's calculate x_n for each n: For n = 0, x_0 = 1. For n = 1, x_1 = (1 + 5 + 6)/6 = 2. For n = 2, x_2 = (8 + 10 + 6)/6 = 24/6 = 4. For n = 3, x_3 = (27 + 15 + 6)/6 = 48/6 = 8. For n = 4, maximum x_4 = (64 + 20 + 6)/6 = 90/6 = 15. Wait, let's check possible values of x_4 given that after the third cut, the pieces are identical. If pieces after the 3rd cut are identical, the cuts are symmetric. The maximum number of pieces with 4 cuts is 15. However, due to symmetry or specific cut positioning, x_4 can take values like 16? Wait! The formula for maximum pieces is (n^3 + 5n + 6)/6, which gives 15 for n=4. But wait, if cuts intersect in certain ways, can x_4 be 16? No, maximum is 15. Wait, let's check option (d) 5. Can x_4 be 5? If 3 cuts produce identical pieces (e.g., 8 pieces), a 4th cut can increase the number of pieces. Since x_3 = 8, a 4th cut can add at least 1 piece, making x_4 >= 9. Therefore, x_4 cannot be 5, because 5 is less than 8 (the number of pieces after 3 cuts). Thus, 5 is not a possible value for x_4.