This work bring **magic cubes** with six sides as magic squares of orders 3 to 10. The **odd order** magic squares are constructed in such a way that they are of **equal difference magic sums**. In each case, the difference is the order of the magic square. The **even order** magic squares are of equal sums. Some examples of **universal** and **upside-down** magic cubes are also given, but this work is given in another site.

These examples are based on different types of magic squares, such as, **single-digit bordered, double-digits bordered, striped, cornered**, etc. The readers can also access the following links to download the whole work.

The readers can also access the following links to download the whole work.

**Inder J. Taneja,**Magic Cubes Based on Magic Squares,**Zenodo**, October 17, 2024, pp. 1-63, https://doi.org/10.5281/zenodo.13924915.- Site Link: Magic Cubes Based on Magic Squares

**Inder J. Taneja,**Universal and Upside-Down Magic Cubes,**Zenodo**, October 17, 2024, pp. 1-38, https://doi.org/10.5281/zenodo.13947425.- Site Link: Universal and Upside-down Magic Cubes

### Magic Cube with Magic Squares of Order 3

**Example 1.**

Let’s consider the following six magic squares of order 3

These magic squares are constructed using sequential numbres from 1 to 54. The magic sums are

S_{3×3}(1):=75, S_{3×3}(2):=78, S_{3×3}(3):=81, S_{3×3}(4):=84, S_{3×3}(5):=87 and S_{3×3}(6):=90.

These magic squares are of equal difference magic sums.

Magic cube is composed with six magic squares representing front and back. ront and back. First three magic squares represent the front part and the last three represent the back part. See below:

### Magic Cubes with Magic Squares of Order 4

**Example 1. **

Let’s consider the following six magic squares of order 4

These magic squares are constructed using sequential numbres from 1 to 96. In this case, all the magic squares are **pandiagonal **and of equal sums, i.e.,

S_{4×4}(1)=S_{4×4}(2)=S_{4×4}(3)=S_{4×4}(4)=S_{4×4}(5)=S_{4×4}(6):=194.

Magic cube is composed with six magic squares representing front and back. ront and back. First three magic squares represent the front part and the last three represent the back part. See below:

**Example 2. **

Let’s consider the following six magic squares of order 4

These magic squares are constructed using sequential numbres from 1 to 96. In this case, all the magic squares are of equal sums, i.e.,

S_{4×4}(1)=S_{4×4}(2)=S_{4×4}(3)=S_{4×4}(4)=S_{4×4}(5)=S_{4×4}(6):=194.

These magic squares are constructed with equal sums **stripes or magic rectangles of order 2×4**.

Magic cube is composed with six magic squares representing front and back. ront and back. First three magic squares represent the front part and the last three represent the back part. See below:

### Magic Cubes with Magic Squares of Order 5

**Example 1. **

Let’s consider the following six magic squares of order 5

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

S_{5×5}(1):=365, S_{5×5}(2):=370, S_{5×5}(3):=375 S_{5×5}(4):=380, S_{5×5}(5):=385 and S_{5×5}(6):=390.

Interestingly, these sums are of equal differences. Each magic square of order 5 is pandiagonal.

**Example 2. **

Let’s consider the following six magic squares of order 5

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

S_{5×5}(1):=365, S_{5×5}(2):=370, S_{5×5}(3):=375 S_{5×5}(4):=380, S_{5×5}(5):=385 and S_{5×5}(6):=390.

Interestingly, these sums are of equal differences. Magic squares of order 5 are **single digit bordred** magic squares.

**Example 3. **

Let’s consider the following six magic squares of order 5

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

S_{5×5}(1):=365, S_{5×5}(2):=370, S_{5×5}(3):=375 S_{5×5}(4):=380, S_{5×5}(5):=385 and S_{5×5}(6):=390.

Interestingly, these sums are of equal differences.

### Magic Cubes with Magic Squares of Order 6

**Example 1. **

Let’s consider the following six magic squares of order 6

These magic squares are constructed using sequential numbres from 1 to 216. These are of equal sums, i.e,

S_{6×6}(1)=S_{6×6}(2)=S_{6×6}(3)=S_{6×6}(4)=S_{6×6}(5)=S_{6×6}(6):=651.

**Example 2. **

Let’s consider the following six magic squares of order 6

These magic squares are constructed using sequential numbres from 1 to 216. These magic squares are bordered magic squares, where the inner blocks are magic square of order 4. These are of equal sums, i.e,

S_{6×6}(1)=S_{6×6}(2)=S_{6×6}(3)=S_{6×6}(4)=S_{6×6}(5)=S_{6×6}(6):=651.

**Example 3. **

Let’s consider the following six magic squares of order 6

These magic squares are constructed using sequential numbres from 1 to 216. These are of equal sums, i.e,

S_{6×6}(1)=S_{6×6}(2)=S_{6×6}(3)=S_{6×6}(4)=S_{6×6}(5)=S_{6×6}(6):=651.

These magic squares are cornered magic squares, where the superior left corners are magic square of order 4.

### Magic Cubes with Magic Squares of Order 7

**Example 1. **

Let’s consider the following six magic squares of order 7

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

S_{7×7}(1):=1015, S_{7×7}(2):=1022, S_{7×7}(3):=1029, S_{7×7}(4):=1036, S_{7×7}(5):=1043 and S_{7×7}(6):=1050.

Interestingly, these sums are of equal differences. Each magic square of order 7 is **pandiagonal**.

**Example 2. **

Let’s consider the following six magic squares of order 7

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

S_{7×7}(1):=1015, S_{7×7}(2):=1022, S_{7×7}(3):=1029, S_{7×7}(4):=1036, S_{7×7}(5):=1043 and S_{7×7}(6):=1050.

Interestingly, these sums are of equal differences. The magic squares of order 7 are **single digit bordered** magic squares.

**Example 3. **

Let’s consider the following six magic squares of order 7

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

S_{7×7}(1):=1015, S_{7×7}(2):=1022, S_{7×7}(3):=1029, S_{7×7}(4):=1036, S_{7×7}(5):=1043 and S_{7×7}(6):=1050.

Interestingly, these sums are of equal differences. The magic squares of order 7 are **double digit bordered** magic squares.

**Example 4. **

Let’s consider the following six magic squares of order 7

These magic squares are constructed using sequential numbres from 1 to 150. The magic sums are

_{7×7}(1):=1015, S_{7×7}(2):=1022, S_{7×7}(3):=1029, S_{7×7}(4):=1036, S_{7×7}(5):=1043 and S_{7×7}(6):=1050.

Interestingly, these sums are of equal differences. The magic squares of order 7 are **cornered magic squares**.

### Magic Cubes with Magic Squares of Order 8

**Example 1. **

Let’s consider the following six magic squares of order 8

These magic squares are constructed using sequential numbres from 1 to 384. These are of equal sums, i.e,

S_{8×8}(1)=S_{6×6}(2)=S_{8×8}(3)=S_{8×8}(4)=S_{8×8}(5)=S_{8×8}(6):=1540.

All the magic squares of order 4 are **pandiagonal** of equal magic sums.

**Example 2. **

Let’s consider the following six magic squares of order 8

These magic squares are constructed using sequential numbres from 1 to 384. These are of equal sums, i.e,

S_{8×8}(1)=S_{6×6}(2)=S_{8×8}(3)=S_{8×8}(4)=S_{8×8}(5)=S_{8×8}(6):=1540.

These magic squares are known **as single digit bordred magic squares**. The inner block of order 4 in each case is a magic square of order 4.

**Example 3. **

Let’s consider the following six magic squares of order 8

These magic squares are constructed using sequential numbres from 1 to 384. These are of equal sums, i.e,

S_{8×8}(1)=S_{6×6}(2)=S_{8×8}(3)=S_{8×8}(4)=S_{8×8}(5)=S_{8×8}(6):=1540.

These magic squares are known as** double digits bordred magic squares**. The inner block of order 4 in each case is a magic square of order 4 with equal magic sums.

**Example 4. **

Let’s consider the following six magic squares of order 8

S_{8×8}(1)=S_{6×6}(2)=S_{8×8}(3)=S_{8×8}(4)=S_{8×8}(5)=S_{8×8}(6):=1540.

These magic squares are constructed using equal sums magic rectangles of order 2×4. These are known by **striped magic squares**.

**Example 5. **

Let’s consider the following six magic squares of order 8

S_{8×8}(1)=S_{6×6}(2)=S_{8×8}(3)=S_{8×8}(4)=S_{8×8}(5)=S_{8×8}(6):=1540.

These magic squares are known by striped magic squares. These are contructed using magic rectangles of orders 2×4 and 2×8.

**Example 6. **

Let’s consider the following six magic squares of order 8

S_{8×8}(1)=S_{6×6}(2)=S_{8×8}(3)=S_{8×8}(4)=S_{8×8}(5)=S_{8×8}(6):=1540.

These magic squares are known as **conered magic squares**. Magic squares of order 4 are at corner of each magic square of order 8. These as cornered with magic squares of order 4 and 6. All the magic squares of order 4 are **pandiagonal** of equal magic sums.

### Magic Cubes with Magic Squares of Order 9

**Example 1. **

Let’s consider the following six magic squares of order 9.

These magic squares are constructed using sequential numbres from 1 to 486. The magic sums are

S_{9×9}(1):=2169, S_{9×9}(2):=2178, S_{9×9}(3):=2187, S_{9×9}(4):=2196, S_{9×9}(5):=2205 and S_{9×9}(6):=2214.

Interestingly, these sums are of equal differences. Magic squares of order 3 are **semi-magic** squares of equal sums in each case.

**Example 2. **

Let’s consider the following six magic squares of order 9.

These magic squares are constructed using sequential numbres from 1 to 486. The magic sums are

S_{9×9}(1):=2169, S_{9×9}(2):=2178, S_{9×9}(3):=2187, S_{9×9}(4):=2196, S_{9×9}(5):=2205 and S_{9×9}(6):=2214.

Interestingly, these sums are of equal differences. Magic squares of order 3 are **magic squares** with different magic sums.

**Example 3. **

Let’s consider the following six magic squares of order 9.

These magic squares are constructed using sequential numbres from 1 to 486. The magic sums are

S_{9×9}(1):=2169, S_{9×9}(2):=2178, S_{9×9}(3):=2187, S_{9×9}(4):=2196, S_{9×9}(5):=2205 and S_{9×9}(6):=2214.

Interestingly, these sums are of equal differences. 9 entries in each block of order 3×3 are of equal sums. The magic squares order 9 are also** bimagic. **

**Example 4. **

Let’s consider the following six magic squares of order 9.

These magic squares are constructed using sequential numbres from 1 to 486. The magic sums are

_{9×9}(1):=2169, S_{9×9}(2):=2178, S_{9×9}(3):=2187, S_{9×9}(4):=2196, S_{9×9}(5):=2205 and S_{9×9}(6):=2214.

Interestingly, these sums are of equal differences. In this case, the magic squares of order 9 are **single digit bordered magic squares**.

**Example 5. **

Let’s consider the following six magic squares of order 9.

These magic squares are constructed using sequential numbres from 1 to 486. The magic sums are

_{9×9}(1):=2169, S_{9×9}(2):=2178, S_{9×9}(3):=2187, S_{9×9}(4):=2196, S_{9×9}(5):=2205 and S_{9×9}(6):=2214.

Interestingly, these sums are of equal differences. In this case, the magic squares of order 9 are **doble digits bordered magic squares**.

**Example 6. **

Let’s consider the following six magic squares of order 9.

These magic squares are constructed using sequential numbres from 1 to 486. The magic sums are

_{9×9}(1):=2169, S_{9×9}(2):=2178, S_{9×9}(3):=2187, S_{9×9}(4):=2196, S_{9×9}(5):=2205 and S_{9×9}(6):=2214.

Interestingly, these sums are of equal differences. Magic squares of order 9 are **cornered magic squares**.

### Magic Cubes with Magic Squares of Order 10

**Example 1. **

Let’s consider the following six magic squares of order 10

These magic squares are constructed using sequential numbres from 1 to 600. These are of equal magic sums, i.e,

S_{10×10}(1)=S_{10×10}(2)=S_{10×10}(3)=S_{10×10}(4)=S_{10×10}(5)=S_{10×10}(6):=3005.

**Example 2. **

Let’s consider the following six magic squares of order 10

These magic squares are constructed using sequential numbres from 1 to 600. These are of equal magic sums, i.e,

S_{10×10}(1)=S_{10×10}(2)=S_{10×10}(3)=S_{10×10}(4)=S_{10×10}(5)=S_{10×10}(6):=3005.

These magic squares are known as **single digit bordered magic squares**. The inner block in each case is a **pandiagonal magic square of order 4.**

**Example 3. **

Let’s consider the following six magic squares of order 10

These magic squares are constructed using sequential numbres from 1 to 600. These are of equal magic sums, i.e,

S_{10×10}(1)=S_{10×10}(2)=S_{10×10}(3)=S_{10×10}(4)=S_{10×10}(5)=S_{10×10}(6):=3005.

These magic squares are also **block-bordered magic squares**. In this case, inner blocks of order 8 are **pandigonal **magic squares of order 8.

**Example 4. **

Let’s consider the following six magic squares of order 10

S_{10×10}(1)=S_{10×10}(2)=S_{10×10}(3)=S_{10×10}(4)=S_{10×10}(5)=S_{10×10}(6):=3005.

These magic squares are also **block-bordered magic squares**. In this case, inner blocks of order 8 are striped magic squares of order 8. The stripes are magic rectangles of order 2×4.

**Example 5. **

Let’s consider the following six magic squares of order 10

S_{10×10}(1)=S_{10×10}(2)=S_{10×10}(3)=S_{10×10}(4)=S_{10×10}(5)=S_{10×10}(6):=3005.

These are double digit bordered magic squares. The inner blocks are magic squares of order 6.

**Example 6. **

Let’s consider the following six magic squares of order 10

S_{10×10}(1)=S_{10×10}(2)=S_{10×10}(3)=S_{10×10}(4)=S_{10×10}(5)=S_{10×10}(6):=3005.

These are cornered magic squares, the corneres are magic squares of order 8, 6 and 10.

## References

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