10. You have 59 cubic blocks. What is the minimum number that needs to be taken away in order to construct a solid cube with none left over?32271816

1. Break Down the Problem

To construct a solid cube, the total number of cubic blocks must be a perfect cube, which can be expressed as $n^3$ for some integer $n$. We need to find the largest $n^3$ that is less than or equal to 59 and then determine how many blocks need to be removed to achieve this.

2. Relevant Concepts

To identify the maximum cube:

• Calculate $n^3$ for different values of $n$.
• Start with $n = 1, 2, 3, \ldots$ until $n^3$ exceeds 59.

3. Analysis and Detail

1. For $n = 1$: $1^3 = 1$
2. For $n = 2$: $2^3 = 8$
3. For $n = 3$: $3^3 = 27$
4. For $n = 4$: $4^3 = 64$ (exceeds 59)

The largest perfect cube ≤ 59 is $3^3 = 27$. To find the number of blocks to remove: $59 - 27 = 32$

4. Verify and Summarize

The calculation indicates that removing 32 blocks will leave us with a perfect cube of 27 blocks.

Final Answer

To construct a solid cube with no blocks left over, a minimum of 32 blocks must be taken away.