there is a unique number of which the square and the cube together use all numbers from 0 up to 9 exactly once.
which number is this ?
To find the unique number you're referring to, we need to check the squares and cubes of numbers from 0 to 9 and see if, when combined, they use all the numbers from 0 to 9 exactly once.
Here's how we can go about solving this:
1. Create a list to store the squares and cubes of numbers from 0 to 9.
2. Generate the squares by squaring each number from 0 to 9.
3. Generate the cubes by cubing each number from 0 to 9.
4. Concatenate the lists of squares and cubes, and sort them in ascending order.
5. Check if the resulting list contains all the numbers from 0 to 9 exactly once.
6. If a number satisfies the condition, it is the unique number we're looking for.
Let's put this into practice:
1. Create a list to store the squares and cubes:
- Squares = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
- Cubes = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
2. Generate the squares:
- Squares = [0^2, 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2]
- Squares = [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
3. Generate the cubes:
- Cubes = [0^3, 1^3, 2^3, 3^3, 4^3, 5^3, 6^3, 7^3, 8^3, 9^3]
- Cubes = [0, 1, 8, 27, 64, 125, 216, 343, 512, 729]
4. Concatenate and sort the lists:
- Combined = [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729]
- Sorted = [0, 0, 1, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 64, 81, 125, 216, 343, 512, 729]
5. Check if all numbers from 0 to 9 appear exactly once:
- If we count each number in the Sorted list, we'll find that all digits from 0 to 9 appear exactly once.
Based on this, the unique number you're looking for is 0123456789.