I almost always take up Ernie’s offers to crew on his boat, when he suggests an overnight expedition. But when he phoned last Thursday I had to remind him that it was TV night, and Acme Satellite Studios were airing the final episode of our favorite Nordic Noir series. But when he assured me that we could watch it on board, I grabbed my wet weather gear and headed down to the docks. Ernie’s boat is rotund, and more than a little scruffy – so much so that the Yacht Club prefer him to park it round the back of the dock, away from their gleaming schooners and cabin cruisers. But it is always seaworthy and Ernie keeps it loaded up with the latest in radar, acoustic depth-meters, ship-to-shore radio, satellite navigation and all sorts of electronic sailing aids. Today there was a new addition – a gimbal-mounted dish antenna bolted to the fore-mast. “It’s controlled by the latest technology, mail-ordered from Acme Industries”, he announced proudly, “and guaranteed to maintain a good television signal, even on a rocking boat!”.
After an enjoyable day exploring the bays and sounds, we made anchor off Stormy Island and secured the boat for the night. As forecast, the wind freshened to the south, so by the time we had cooked dinner and settled down in front of the newly-installed 32’’ TV, the boat was rocking and pitching somewhat. All was going well until the last five minutes of the show when, without warning, the screen went blank and only white noise could be heard from the speakers. Ernie fiddled with the settings, but it was a good ten minutes before we got it working again. By then it was too late – the program had finished. I don’t think I have ever seen Ernie so angry. It took all my powers of persuasion to stop him ripping out the dish controller and hurling it off the boat! We didn’t find out what Saga would finally do with her suspicions about Martin until a belated breakfast at my place after we sailed back the next morning (luckily I had recorded the program at home).
After breakfast Ernie began to disassemble the satellite receiver on the kitchen table. “Good grief, I’ve been scammed!”, he cried. “Look at this!”. He pointed at the Acme Industries label on the front of the box spot which was beginning to peel off. Underneath was a second label reading Acne Imdustries. “It’s a cheap rip-off from Kzijekistan”, he said, and explored more deeply into the innards of the device. “To be fair”, he continued, “most of it is reasonably well made, but the optical gyroscopes are very third rate” - and he pulled out three small devices and displayed one of them on his palm. It was a pentagonal prism made of transparent plastic with a hole bored along its length and some optics mounted at one end. Ernie drew a diagram on the kitchen white-board to explain how it was supposed to work, but I’m afraid it went a bit over my head (I took a snapshot with my phone-camera for so I could study it more carefully later).
“I can see why they didn’t work properly”, he announced. “The silver paint on the outside is coming off in patches, and the superconducting film on the inside is much too thin – but I think the real problem is the diameter of the bore-holes – they’re obviously too small for efficient signal detection. But I think that with a little work we might be able to get things working satisfactorily.” Ernie instructed me to drill the hole out a bit in one of the prisms while he made a few ‘improvements’ to the electronics. “What diameter do you want?” I asked. But Ernie wasn’t quite sure - “Oh, just make it a just a little bit larger to start with – we can always make the hole bigger if we need to”. So I chose a drill at random, carefully enlarged the hole, touched up the outside (Acme Silver Mirror paint), and carefully re-applied the detector layer (Acme Room-temperature Superconducting paint).
When the paint was dry Ernie connected the reconfigured gyroscope to his 'portable-pocket-meter' with a flexible lead and tested it by spinning it between his palms. “Extraordinary”, he muttered, “I couldn’t have asked for better performance! You appear to have hit the sweet spot with the diameter of the hole you drilled”. Ernie explained that from the shape of the signal he could tell that at least some of the radial beam was bouncing off five external prism walls before returning to the detector cylinder light to bounce five times off the outside walls of the prism before it hit the detector, but none was bouncing six times or more. “Tell you what...”, he continued “I’ve got to go now, business in town, but why don’t you drill out the holes in the other two detectors to the same size, paint them up just like this one, and plug them back into the box? Then I can drop by next Thursday afternoon, pick you up, reinstall the satellite system on the boat and we can watch Season 3, Episode 1, anchored off Hurricane Point on Thursday evening”. And in his usual abrupt manner, he left the house without waiting for my reply.
Now I have just one problem. While waiting for the paint to dry I tidied up and put the drill bit back on the rack without checking which one it was. The drills were a gift from Ernie and go up in 1 micron intervals all the way from 0.1 mm to 12.5 mm (it’s a big set), and I just can’t remember which one I used. I have checked that the side length of each prism is exactly 10 mm. Can anyone help by telling me the minimum and maximum diameters I can drill to ensure at least some of the light will undergo at least, but no more than, five reflections?