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   A magical number?
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   Author  Topic: A magical number?  (Read 698 times)
BMAD
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A magical number?  
« on: Feb 24th, 2015, 9:01am »
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Write a list from 1 to 100.
Pick two numbers at random
Sum the two numbers and find the product of the two numbers
Sum the above two numbers you found
Erase the two chosen numbers from the list
Add the final sum of the two numbers to the list
Repeat until there is only one number.
 
Repeat the above process multiple times.  
 
Is it the same number? Do the numbers center around a point? Bimodal?  analyze.
 
I recommend trying this for numbers 1-10 and 1-20 to see if anything special is happening ... The calculated numbers can be quite huge.
« Last Edit: Feb 24th, 2015, 9:02am by BMAD » IP Logged
towr
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Re: A magical number?  
« Reply #1 on: Feb 24th, 2015, 11:15am »
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The operation is commutative and associative, so the answer is always the same for a given starting set of numbers
It seems to be to sum of the products of each set in the powerset. But calculated much quicker than it sounds :p
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pex
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Re: A magical number?  
« Reply #2 on: Feb 24th, 2015, 2:32pm »
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Numerically, the result seems to be (n+1)!-1. Given the form of the answer and towr's observation that the order in which the operations are performed doesn't matter, it looks like it shouldn't be too hard to produce a proof based on good old mathematical induction, but I don't have the time right now.
 
Edit: Ok, the base case is obvious, and the induction step is essentially just (n!-1) + n + n(n!-1) = n!(n+1) +n-n -1 = (n+1)!-1. Neat little puzzle!
 
Edit the Second: By the same reasoning, if the list of numbers does not have to be 1..n but can be anything, the result is product(xi+1)-1.
« Last Edit: Feb 25th, 2015, 1:38pm by pex » IP Logged
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