Behold the solution to The Wright Challenge: Beyond the Puzzledome #3. If you
forgot the actual question, click here to see it
again.
Any answer of the form 253 + 256k will work for the coconut problem. The most realistic answer of those would be 253, but we accepted 509, 765, 1018 or any of that ilk. One group pointed out that -3 works, and that the travelers OWE coconuts. Doctor E loves his contestants.
Now... how do you GET that result?
Here is one way. Let N be the total amount of coconuts, N1 be the amount the first traveler leaves, N2 be the amount the second traveler leaves, etc. We start by looking at the amount the last person left, N4. If the last person left N4, the penultimate person left N3 where N3 = 4N4/3 + 1 or (4N4 + 3)/3 . Then the antepenultimate person left N2 = (4N3 + 3)/3 = (4(4N4 + 3)+9)/9. Then the preantepenultimate person left N1 = (4N2 + 3)/3 = (4(4(4N4 + 3) + 9 ) + 27) / 27 And the original pile contained N = (4N1 + 3)/3 = 4(4(4(4(N4 + 3) + 9) + 27) + 81)/81 = (256N4 + 525)/81 coconuts. So we need to find a value of N4 such that (256N4 + 525)/81 is an integer.
Read this part if you understand Number Theory:
We can find permissible values of N this way:
Read this part if you are unfamiliar with Number Theory:
We need to find a value of N4 such that (256N4 + 525)/81 is an integer. In other words, we want 256N4 + 525 to be a multiple of 81, that is, to have a remainder of zero when we divide by 81. Since 525/81 gives a remainder of 39, we want 256N4 to give a remainder of (81-39 = 42) when we divide by 81. Our arithmetic can be simplified if we realize that 256 = (81 * 3 + 13), so we are just looking for an N4 such that 13N4 gives a remainder of 39 when we divide by 81. This is tedious, but not overly so, and we wind up with 78.
Click here for Doctor E's special message to YOU!
The etymology of the word orange is quite interesting. It was of Dravidian origin (a group language spoken in and
around India), and eventually adopted into Sanskrit as "narangah." It was pronounced Nar-an-hah, as the current day Spanish pronounce it. (The spanish word is naranja). Now the English speakers saw "narang" "Naranj" etc. and started pronouncing it like it was spelled: Nar-an-je. What happened to the beginning N? Glad you asked. People would say things like "Please give me a narange" and then it was slurred to "Please give me anarange" and eventually "Please give me an arange" The N migrated to the indefinite article!
This happened a lot in our language. The word "apron" used to be "napron" but it lost its N the same way. The word "Newt" used to be "Ewt" but it GAINED an N by that process.
You have just read the solution to The Wright Challenge: Beyond the Puzzledome #3. Please feel free to email us your comments and feedback. Why not try the current Wright Challenge now?
Web Page design: Doug Shaw
Solutions can be emailed to doctore@math.uni.edu, dropped off at the math office (220 Wright Hall), or mailed to Doctor E, c/o University of Northern Iowa mathematics department, 220 Wright Hall, Cedar Falls, IA 50614. Please write neatly and clearly, including your name and address (if you are competing as an individual) or your team name with the names of its members, and a contact address. If there are any questions you can email doctore@math.uni.edu. The solution will be posted on this website after it is due.
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Beyond the Puzzledome #1
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to go to the solution to Beyond the Puzzledome #1
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The Wright Challenge, Fall 1999
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Wright Challenge: Doctor E's Revenge, Spring 2000
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Wright Challenge: Doctor E Strikes Back, Fall 2000
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Wright Challenge: Doctor E Returns, Spring 2001
Web Page design: Doug Shaw