I am building a mechanical clock that is intended to show the position of the visible planets and the phase of the moon over the next ten thousand years. Thus, I would like a gear train that computes all the following ratios using a small number of gears. 29.53058853, 87.96925718, 224.7007992, 365.2563554, 686.9795859, 4332.820129, 10755.69864 The gear train must use gears with tooth number not less than 12 and not greater that 400. I want to integrate daily rotations to show the orbital position within a few percent after 10,000 years, so I need accuracy to at least 6 places. After playing around with this problem using lattice methods, I realized that (a) there were like some very good solutions in the space and (b) that Bill Gosper would probably be better at finding them than I would. Indeed, RWG took up the challenge and kindly suggested the following solution (note the shared gears): RWG wrote:

293 149 157 (--- ---) --- = 29.5350888532671 +.3 lsd 12 83 233

293 152 358 101 61 (--- --- ---) --- -- = 87.9692571945795 1.4 lsd 12 107 295 67 44

293 152 149 (---) --- --- = 224.700799133144 -.7 lsd 12 107 23

(Don't see how to reuse 149.)

293 152 358 269 (--- ---) --- --- = 365.256355463807 +.6 lsd 12 107 295 31

293 152 358 101 263 267 (--- --- --- ---) --- --- = 686.979585832448 -.7 lsd 12 107 295 67 47 138

293 152 358 101 263 322 281 272 (--- --- --- --- --- ---) --- --- = 4332.8201300432 1 lsd 12 107 295 67 47 103 107 183

(Don't see how to reuse 281.)

293 152 358 101 263 322 281 (--- --- --- --- ---) --- --- = 10755.6986519239 1.2 lsd 12 107 295 67 47 103 29

Beat that! --Bill

I post his solution here because it is lovely and also in the faint hope that someone on the list may be able to do better. I offer a $1 prize to the first person (including Bill) who can beat this solution, by reducing the number of gears size without decreasing the worst-case accuracy. -Danny