To find the odd numbers that sum up to a given cube, we can use a property of cubes: the sum of the first $n$ odd numbers is equal to $n^2$. Hence, the sum of the first $n^2$ odd numbers is equal to $n^3$. Using this property, we can find the odd numbers that sum up to the given cubes.

(a) $7^3$

To find the odd numbers that sum to $7^3 = 343$:

7^3 = 343 \quad \text{so, we need the first 7^2 = 49 odd numbers.}

The first 49 odd numbers are: $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97$

Sum of these 49 odd numbers is $343$.

(b) $12^3$

To find the odd numbers that sum to $12^3 = 1728$:

12^3 = 1728 \quad \text{so, we need the first 12^2 = 144 odd numbers.}

The first 144 odd numbers are: $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297$

Sum of these 144 odd numbers is $1728$.

(c) $4^3$

To find the odd numbers that sum to $4^3 = 64$:

4^3 = 64 \quad \text{so, we need the first 4^2 = 16 odd numbers.}

The first 16 odd numbers are: $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31$

Sum of these 16 odd numbers is $64$.

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Here are some related questions that expand on this information:

1. How can you prove that the sum of the first $n$ odd numbers is $n^2$?
2. What is the formula for the sum of the first $n$ odd numbers?
3. Can you derive the sum of odd numbers for any other cube values?
4. How can you find the nth odd number?
5. What is the significance of the sum of odd numbers in number theory?
6. How do the properties of odd and even numbers differ?
7. How can you visualize the sum of odd numbers geometrically?
8. What are some practical applications of the sum of odd numbers?

Tip: When working with sequences, it’s often helpful to look for patterns and relationships that simplify calculations.