To solve this problem, we need to follow a step-by-step approach, calculating the volume of both the suitcase and the packing cubes, and then determining how many small cubes can fit in the remaining space.

Step 1: Calculate the volume of the suitcase

The suitcase has inside dimensions of 29 inches by 18 inches by 6 inches. The volume of the suitcase is: $\text{Volume of suitcase} = 29 \times 18 \times 6 = 3132 \text{ cubic inches}.$

Step 2: Calculate the volume of one large packing cube

Each large packing cube has dimensions 6 inches thick, 6 inches wide, and 12 inches long. The volume of one large packing cube is: $\text{Volume of one large cube} = 6 \times 6 \times 12 = 432 \text{ cubic inches}.$

Step 3: Calculate the volume of 5 large packing cubes

If you pack 5 large packing cubes, the total volume they occupy is: $\text{Volume of 5 large cubes} = 5 \times 432 = 2160 \text{ cubic inches}.$

Step 4: Calculate the remaining volume in the suitcase

The remaining volume in the suitcase after packing the 5 large cubes is: $\text{Remaining volume} = 3132 - 2160 = 972 \text{ cubic inches}.$

Step 5: Calculate the volume of one small packing cube

Each small packing cube has dimensions 8 inches thick, 4 inches wide, and 4 inches tall. The volume of one small packing cube is: $\text{Volume of one small cube} = 8 \times 4 \times 4 = 128 \text{ cubic inches}.$

Step 6: Determine how many small cubes can fit in the remaining space

To find how many small cubes can fit in the remaining volume, we divide the remaining volume by the volume of one small cube: $\text{Number of small cubes} = \frac{972}{128} \approx 7.59.$ Since we cannot fit a fraction of a cube, we round down to the nearest whole number, which is 7.

Conclusion:

You can fit 7 small packing cubes in the remaining space of your suitcase.

---

Do you need any further details or explanations on this problem?

Here are 5 related questions to explore:

1. How much total volume do the large and small packing cubes occupy together?
2. What if the suitcase had dimensions 30 x 19 x 6 inches—how would this change the outcome?
3. How many large cubes could fit in the suitcase without any small cubes?
4. What would happen if the suitcase height were reduced by 1 inch?
5. How much unused space remains in the suitcase after packing both the large and small cubes?

Tip: When dealing with packing problems, always focus on the shape of the objects and see if rearranging can maximize space!