Methods for making (2n,2)c/x ships in an adjustable slope ships ruleThere are a few ways to make adjustable slope ships in various rules, but for minimum population, the rule B2ae3acnqy4aint5aj6c7e8/S01e2ce3cjnqr4acejknr5-jkqr6ik7e seems to be a good candidate.

To make many of the speeds, we want this base ship, a (2,2)c/20:

`x = 20, y = 21, rule = B2ae3acnqy4aint5aj6c7e8/S01e2ce3cjnqr4acejknr5-jkqr6ik7e`

4$6bo$5bo6bo$4bo8bo$6bo4bo2bo5$2bo2bo$4bo6bo$3bo8bo$13bo$11bo!

Constructing other diagonal speeds from here (of the form (2,2)c/(2n+18), where n>=1) is easy. Simply move the left two sections left one cell, and the bottom two down one cell to get a (2,2)c/22:

`x = 61, y = 14, rule = B2ae3acnqy4aint5aj6c7e8/S01e2ce3cjnqr4acejknr5-jkqr6ik7e`

4bo46bo$3bo6bo22bo16bo7bo$2bo8bo37bo9bo$4bo4bo2bo22bo15bo5bo2bo2$28bob

obo4bo2$35bo$o2bo$2bo6bo23bo13bo2bo$bo8bo38bo7bo$11bo36bo9bo$9bo49bo$

57bo!

Using this base ship, we can make ships with speeds of the form (4m-2,2)c/(36m-18+4mn-2n) for m>=1. By taking the base ship and moving the bottom two parts down 20 cells, we get a ship of slope (6,2). For general diagonal ships of speed (2,2)c/(2n+18), to get a slope (6,2) ship, move the bottom bits down 18+2n cells. This operation changes a (2,2)c/22 ship into a (6,2)c/66 ship, and a (2,2)c/20 ship into a (6,2)c/60:

`x = 42, y = 66, rule = B2ae3acnqy4aint5aj6c7e8/S01e2ce3cjnqr4acejknr5-jkqr6ik7e`

4bo28bo$3bo7bo20bo6bo$2bo9bo18bo8bo$4bo5bo2bo19bo4bo2bo5$29bo2bo$o2bo

27bo6bo$2bo7bo19bo8bo$bo9bo28bo$12bo25bo$10bo17$4bo28bo$3bo7bo20bo6bo$

2bo9bo18bo8bo$4bo5bo2bo19bo4bo2bo25$29bo2bo$31bo6bo$30bo8bo$o2bo36bo$

2bo7bo27bo$bo9bo$12bo$10bo!

If you want to increase n for higher slope ships, instead of moving the left two bits one cell left and the bottom two one cell down, move the left two one cell left and the bottom two 2m-1 cells down, where the m is the m in the general formula (4m-2,2)c/(36m-18+4mn-2n). Here is the transformation from (10,2)c/100 to (10,2)c/110:

`x = 56, y = 58, rule = B2ae3acnqy4aint5aj6c7e8/S01e2ce3cjnqr4acejknr5-jkqr6ik7e`

4bo41bo$3bo6bo34bo7bo$2bo8bo32bo9bo$4bo4bo2bo33bo5bo2bo22$24bo2$26bo2$

21bobobo2bo2$26bo2$24bo15$o2bo$2bo6bo$bo8bo$11bo$9bo$42bo2bo$44bo7bo$

43bo9bo$54bo$52bo!

Using the steps above, it is possible to construct any ship of a speed of the form (4m-2,2)c/(36m-18+4mn-2n) in only 16 cells. There is a way to construct some other periods, but that will be in the next post.