251. Packing Cylinders. Forty cylinders with a diameter of 1 cm and the same height were tightly packed in a box in 5 rows of 8 cylinders each so that they would not "rattle" during transportation. How many cylinders need to be removed from the box so that, by moving the remaining cylinders and adding the removed cylinders and one more cylinder at the end, 41 cylinders of the same size can be packed in this box? Will the cylinders "rattle" in this case?

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-054.jpg?height=325&width=512&top_left_y=1448&top_left_x=772)
251. Only two cylinders need to be removed from the box. Let's renumber the cylinders as shown in the figure. Remove cylinders 6 and 16.

Move cylinders $7-10$ to the right and up (see the figure). Move cylinders 11-15 to the left, cylinders 17-20 up and to the left, and cylinders 21-25 to the left. The remaining cylinders can be moved in different ways. For example, move cylinders 26, 28, 29, 30, 31 to the left; press cylinder 32 against cylinders 31 and 37; move cylinder 27 against cylinders 28 and 32; move cylinder 26 against cylinders 21 and 22. Move cylinder 31 to the left; press cylinder 32 against cylinders 36 and 37; move cylinder 27 up to cylinder 31; move cylinders $28, 29, 30$ up and to the left; move cylinders $33, 34, 35$ up and to the right. Insert cylinders 6, 16, and 41 into the positions indicated in the figure. If the cylinders in the new arrangement are tightly pressed against each other, the distance between the lines of centers of adjacent "columns" will be $\sqrt{\frac{3}{2}}$ cm. Therefore, the entire packaging will take up a length of $8\left(\frac{\sqrt{3}}{2}\right)+1=4 \sqrt{3}+1 \approx 7.928 < 8$ cm.

Thus, the cylinders will now "rattle" slightly during transportation.