What are the maximum possible number of pieces that can be cut from a large cube, by doing only 15 cuts?

Question

What are the maximum possible number of pieces that can be cut from a large cube, by doing only 15 cuts?

1. A. 216 (Correct answer)
2. B. 125
3. C. 225
4. D. 150

Correct Answer

Option A — 216

Detailed Solution & Explanation

The correct answer is 216.

Key Points

• Cuts along a single axis are independent of cuts along the other two, so if the 15 cuts are split as x, y and z across the three perpendicular directions, the number of pieces is (x + 1)(y + 1)(z + 1).
• To maximise a product with a fixed sum, make the factors as equal as possible.
• Splitting 15 as 5 + 5 + 5 gives (5 + 1)(5 + 1)(5 + 1) = 6 × 6 × 6 = 216 pieces.

Additional Information

• Any uneven split does worse — for example 6 + 5 + 4 gives 7 × 6 × 5 = 210, and 15 + 0 + 0 gives only 16.
• This is the discrete form of the general principle that, for a fixed sum, a product is largest when its terms are equal.

*Ministry of Papers worked solution. The Commission has not yet published the official key for this paper; this answer will be reconciled with it on release.*