Since the balls and boxes are indistinguishable, the problem involves finding how many ways we can partition 4 identical balls into 4 identical boxes. This is a combinatorial problem and can be solved using the concept of integer partitions.

Step-by-step solution:

1. We need to partition the integer 4 into at most 4 parts (each part representing the number of balls in each box). The parts must be non-increasing to avoid double-counting the arrangements.

2. The possible partitions of 4 (in non-increasing order) are:

   • $4$: All 4 balls in one box.
   • $3 + 1$: 3 balls in one box, 1 in another.
   • $2 + 2$: 2 balls in one box, 2 in another.
   • $2 + 1 + 1$: 2 balls in one box, and 1 ball each in two other boxes.
   • $1 + 1 + 1 + 1$: Each box gets 1 ball.

3. Let's count these partitions:

   • $4$: 1 way.
   • $3 + 1$: 1 way.
   • $2 + 2$: 1 way.
   • $2 + 1 + 1$: 1 way.
   • $1 + 1 + 1 + 1$: 1 way.

4. Summing up these possibilities gives us:

   $1 + 1 + 1 + 1 + 1 = 5$

So, there are 5 ways to place 4 indistinguishable balls into 4 indistinguishable boxes.

Would you like more details on any part of this solution or have any questions?

Here are some related questions to further your understanding:

1. In how many ways can 5 indistinguishable balls be placed in 3 indistinguishable boxes?
2. What if the balls are distinguishable but the boxes are not? How does the answer change?
3. How many ways can you place 3 indistinguishable balls into 3 distinguishable boxes?
4. How does the solution change if we allow some boxes to be empty?
5. Can you generalize a formula for distributing $n$ indistinguishable balls into $m$ indistinguishable boxes?

Tip: When dealing with problems involving indistinguishable objects, always consider the partitions of the total number to determine possible arrangements.