"Have a look at this" said Ernie when he dropped in unannounced yesterday, pulling what looked like an old-fashioned photographic negative from an envelope. I held the small plastic square up to the light and peered at it, but could see nothing except the grey glossy surface. But after a few seconds later it appeared to turn completely transparent so I could see the light fitting through it. "Try the other side", he suggested. I did, and almost dropped it in surprise as I saw an eye staring back at me. A few seconds the eye image faded to a view of the other side of the room. "It's a new ultra-high refractive index plastic from Acme Industries" said Ernie. "They call it slow glass". He went on to explain that the speed of light through a transparent material was inversely proportional to its refractive index. "...and this material has a refractive index of approximately 10^18". I must have looked confused so, putting on his 'explain-this-to-a-small-child' face he continued "So light hitting one side of a surface coating only a manometer thick would take almost 4 seconds to get out the other side".
"Interesting", I said, "but I can't imagine it being very useful. Just consider the risk of accident if you used it in a car windscreen". But Ernie assured me that while previously it had been a 'solution waiting for a problem', he had now identified the perfect problem. "As you know", he announced (I didn't), "there is a lunar eclipse visible from here in just a few days. Two weeks later there will be a solar eclipse visible in the Kzijekistan hinterlands. I would love to see both but it will take a 16 day trek to get to the site of the solar eclipse and set up my equipment." I murmured some commiserations, but Ernie wasn't finished. "Don't you see - the slow glass is perfect. If someone" (he looked pointedly at me) "exposes a 1.0 mm thick sheet to the lunar eclipse the light will take 40 days to get through so I will have plenty of time to get back and make all the required observations".
He assured me I would need no astronomical abilities (just nimble fingers). "My attic sky-light is perfectly oriented for a great view of the eclipse. All you need to do is tape slow glass sheets up to the sky-light during the eclipse, then store them safely for my return". He went on to explain that as slow glass was a 'very new thing' he didn't quite know the best size or exposure times for optimal eclipse viewing. So he had come up with a cleverly designed compromise involving multiple sheets of slow glass and a range of different exposure times. "It's all very simple", he announced passing an envelope to me, "I have writen down explicit instructions for you". And then he looked at his watch, told me he had to rush and finish packing for the solar eclipse trip, and left with a cheerful farewell.
As soon as he left I opened the envelope to read his instructions. The contents consisted 16 statements, each typed on a separate scrap of paper, plus a scrawled note. In the order I removed them from the envelope, the statements read:
- The area of the largest sheet of slow glass should be precisely 1/4 the area of the sky-light.
- The eclipse will last for precisely 210 minutes.
- The smallest sheet of slow glass should be as small as possible consistent with the rest of these instructions.
- The exposure time for the largest sheet of slow glass should be precisely half that of the eclipse.
- No sheet of slow glass may overlap the edges of the window during its exposure.
- The sky-light is precisely 1000 mm square.
- Each sheet may be exposed at any time, and at any position and orientation on the window that is consistent with the rest of these instructions.
- The exposure time for the smallest sheet of slow glass should be precisely half that of the second to smallest sheet.
- Each sheet of slow glass, except for the smallest and the largest should have exposure time exactly equal to the average of the exposure times of the next smaller and next larger sheets.
- Each slow glass sheet should be exactly square.
- Every part of the eclipse should be recorded by the exposure of at least one sheet of slow glass..
- Each sheet of slow glass, except for the smallest and the largest should have side length exactly equal to the average of the side lengths of the next smaller and next larger sheets.
- Each sheet's exposure must begin at or after the start of the eclipse and end at or before the end of the eclipse.
- Each slow glass sheet will be exposed by taping it flat to the window, without movement, for its entire exposure time.
- The area of the smallest sheet of slow glass should be precisely 1/4 that of the second to smallest sheet.
- No sheet of slow glass may over-lap or under-lap any other sheet during its exposure.
The note read: Sorry, but I haven't had time to buy the slow glass. If you order it for me from Acme Industries I will compensate you upon my return. Note that Acme will only supply square sheets of slow glass - no rectangles supplied. While their price per square meter is reasonable, they have an exorbitant cutting and handling charge for multi-piece orders. So please just buy a single square piece that is just big enough to cut out all the bits you will need - you can cut it up into the required squares yourself using a heated knife.
I am sure you can see my problem. I don't want to disappoint Ernie by buying too much, or too little, or cutting too many or too few pieces, or exposing them for the wrong time. How big a square piece of slow glass should I order? A cutting diagram and some sort of diagram or schedule for exposing the sheets would also be of great help to me.
Clue 1. If you think of time as a third dimension, what shape does a square sheet of slow glass plus its exposure time (or the sky-light plus the time of the eclipse) make?
Clue 2. (Taking Xoff's comments into account) Just consider where you would put the 9-th largest cube...