Archive > The Amazing Race 16 Spoilers/Speculation
TAR16 - EP6: "Cathy Drone?" (France)
apskip:
I see the glass stacking/champagne pouring exercise as foremost a mathematical challenge. If I were running the race and had that task, the first thing I would ask for is calculator (even though I did not use on in what I describe below), pencil and paper tp figure out the theoretically strongest arrangement of glasses. I did this and it took me 12 minutes to get the soundest mathematical solution. My sense is that there are multiple solutions that work but one is probably optimum. Can you figure out the optimum without outside assistance and how long did it take you?
There is much more in this task than just the mathematical solution since how well the structure supports itself depends on how good you are building it (Boston Rob would excel at this part) but since I might have a half dozen champagne glasses available I don't expect to test my mathematical solution any time soon.
chill_sd:
My guess would be a tetrahedral shape, rather than a pyramid. The base would be a triangle with 15 glasses on each side, then 14, 13, etc. This would total 681 680 glasses. Each glass would rest on 3 below it, providing greater stability.
Edit: After reviewing the episode, I thought I would have to leave out one glass to make it 680, but I discovered I had miscalculated one of the levels, so the total of a triangular pyramid with 15 glasses at the base is 680, as stated by Phil.
walkingpneumonia:
--- Quote from: chill_sd on March 22, 2010, 03:37:05 PM ---My guess would be a tetrahedral shape, rather than a pyramid. The base would be a triangle with 15 glasses on each side, then 14, 13, etc. This would total 681 glasses. Each glass would rest on 3 below it, providing greater stability.
--- End quote ---
The example pyramid created by the "sommeliers" has a triangular base, whereas both sets of racers created theirs with a square base. Unfortunately neither team had the benefit of seeing the example and couldn't think outside the box.
I'm still having problems with the math though - how would you calculate this other than drawing it out level by level?
Belle Book:
I would've gone with the grapes task. Sure, it's a luck-based task but I don't think I'd have the skill to do the glasses one. In most cases, it's better to go with skill than luck but there are exceptions -- and this is one of them.
Belle Book
chill_sd:
--- Quote from: walkingpneumonia on March 22, 2010, 03:57:43 PM ---I'm still having problems with the math though - how would you calculate this other than drawing it out level by level?
--- End quote ---
There are some shortcuts for adding up triangular numbers, but as you can see from the edit above, I made a small mistake. When I went back, I used a spread sheet to count all the glasses that would be in each row.
A quick explanation of the shortcut: The bottom level would have rows of glasses starting with 1 at the top of the triangle and progressing to 15 at the bottom. The top and bottom rows add up to 16, as do the 2nd and 14th rows. So you have 7 pairs of rows adding up to 16 glasses each pair, and 8 in the middle row. 7 times 16 plus 8 = 120 glasses in the bottom layer.
PS - I would have used a similar trick to add up the numbers on a clock face (I don't remember which season, but it was part of a task involving finding numbers to open a safe). The clock has 12 numbers, which can be paired up 1-12, 2-11, 3-10, 4-9. 5-8 and 6-7. Six pairs that add up to 13. 6 times 13 = 78, which is the sum of the numbers 1 through 12.
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