Let D be the number of drum lamps needed. This gives us D/5 groups of 5 drum lamps, to produce D/5 pairs of lampions—2D/5 lampions altogether.
The lampions can be grouped into fives, that’s 2D/25 groups, producing 2D/25 pairs of trios—4D/25 trios altogether.
The trios can be grouped into fives: 4D/125 groups, producing 4D/125 lotus pairs, 8D/125 lotus lamps.
Similarly grouping lotus lamps, we get 8D/625 groups to produce 8D/625 pairs of jade turtle, 16D/625 altogether.
Grouping jade turtle lamps, we get 16D/3125 groups, for 16D/3125 pairs of festive carp lamps. If we started with D=3125, we would have 16 pairs of festive carp lamps at the end of the line, or 32 individual festive carp lamps. The final ratio of festive carp to drum lamps is 32:3125, and the final ratio of a pair of festive carp to drum lamps is 16:3125. That is respectively 1:97.65625 and 1:195.3125. This depends on the merging process to permit breaking down the lamp products into smaller components. Under these conditions we would need 97²¹⁄₃₂ drum lamps to produce one festive carp lamp, and 195⁵⁄₁₆ drum lamps to produce 2 festive carp lamps.
[If it takes 5 drum lamps to make 2 lampions, then it takes 2½ drum lamps to make 1 lampion. Assuming that none of the lamp products can be fractional, 3 drum lamps would be needed to make 1 lampion, because 2 would be insufficient. However, this means that the ratio 5:2 cannot be maintained accurately. Let’s see what would happen if we started with 200 drum lamps. Grouped into fives, this would be 40 groups producing 40 pairs of lampions, which is 80 lampions. That’s 16 groups of lampions, producing 16 pairs of trios, 32 trios in all. If we discard 2 trios, we have 30 trios, which is 6 groups producing 12 lotus lamps. Discard 2 of these and we have 10 lotus lamps, or two groups of lotus lamps, producing two pairs, or 4 jade turtle lamps, insufficient to make 2 festive carp lamps, but enough to make 1 festive carp lamp, because 4>2½. The process can be represented by a 6-term series (drum, lampion, trio, lotus, jade, carp)=(200, 80, 32, 12, 4, 1). Starting with a number of drum lamps we can see how many festive carp lamps we end up with, discarding any lamp products we can’t group along the way.
Only by starting with 3125 drum lamps can we maintain the fixed ratio 5:2 and end up with 16 pairs of festive carp lamps.
Specifically, if a jade turtle lamp is capable of splitting into two for the purposes of merging, but all other types cannot be split, You can merge between 165 and 249 drum lamps to produce 1 festive carp lamp; and you can merge between 250 and 414 drum lamps to produce 2 (or maybe 3) festive carp lamps. Use the series explained above to group the various types of lamp.]