20201201, 07:19  #1 
Jul 2015
26_{10} Posts 
December 2020
Last fiddled with by LaurV on 20201201 at 07:21 Reason: fixed link 
20201201, 07:41  #2 
Jul 2015
2·13 Posts 
Error in example?
134 160 206 235 265 = 1000 and not 1001
I think the correct numnber should be 134 161 206 235 265 Last fiddled with by tgan on 20201201 at 07:43 
20201201, 09:27  #3 
Jul 2015
11010_{2} Posts 

20201201, 09:58  #4 
Romulan Interpreter
Jun 2011
Thailand
3×47×67 Posts 
Yep, you are right on both counts. They will probably correct it later.
I fixed your first post (runaway link). 
20201201, 12:33  #5 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
6,143 Posts 
I count 67 unique solutions.
Have puzzles in the past had multiple solutions, or did I just make a mistake somewhere? 
20201201, 14:56  #6  
Jul 2015
2×13 Posts 
Quote:
must admit that i did not totally understood the puzzle. do we look for a maximum or we need to get the required number? Last fiddled with by tgan on 20201201 at 14:57 Reason: spelling 

20201201, 15:24  #7 
"Ed Hall"
Dec 2009
Adirondack Mtns
3,719 Posts 
I remember past puzzles with many answers. A few were "special" and gained extra credit. Special, such that a name may appear or a palindromic answer, etc. within the solutions.

20201201, 15:44  #8 
Feb 2017
Nowhere
2^{6}×71 Posts 
I don't see anything about "the" solution being unique, so why not multiple solutions? I'm not sure exactly what "67 'unique' solutions" means; it seems like an oxymoron. I'm guessing "unique" refers to a specific ordering of the 5 population numbers, so you don't have two solutions with the same five population numbers.
Nonuniqueness of solutions is no major defect, but geez  they can't even give a proper example. Pathetic. Struggling to derive an interesting puzzle from the conditions... It occurred to me to wonder how many ways there are of expressing 1001 as the sum of 5 odd positive integers. Subtracting 1 from each summand gives 5 nonnegative even integers. Dividing through by 2, we have the the problem of expressing 498 as the sum of 5 nonnegative integers. This is wellknown to be equal to the number of ways of expressing 498 as the sum 498 = x_{1} + 2*x_{2} + 3*x_{3} + 4*x_{4} + 5*x_{5} where the x's are nonnegative integers; the corresponding 5 summands of 498 are x_{5}, x_{5} + x_{4}, x_{5} + x_{4} + x_{3}, x_{5} + x_{4} + x_{3} + x_{2}, and x_{5} + x_{4} + x_{3} + x_{2} + x_{1}. The number of such 5tuples is approximately the volume of the 5dimensional "simplex" in Euclidean 5space bounded by the coordinate axes and the hyperplane given by the above equation. This volume is 498^5/(5!*5!), which is approximately 2,000,000,000. Last fiddled with by Dr Sardonicus on 20201201 at 16:14 Reason: Forgot extra factor of 5! in denominator 
20201201, 22:28  #9  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
6,143 Posts 
Quote:
I only included odd population numbers, as the puzzle says. But the example has even numbers. So I'm not sure what to make of that. It all seems a bit muddled with the errors and nonconforming examples. Last fiddled with by retina on 20201201 at 22:29 

20201202, 00:39  #10 
Feb 2017
Nowhere
2^{6}×71 Posts 
Upon rereading the problem, I find I hadn't been reading it correctly. I think I have it now, but if so I have a major difficulty with it.
A component p_{i} of the vector pop is the numbers of eligible voters in state i. The hypothesis that all the v_{i} are odd insures that there can't be a tie at the ballot box in any state. The corresponding component v_{i} of the vote is the number of votes a candidate got in state i. Therefore v_{i} <= p_{i}, and if 2*v_{i} < p_{i}, then the other candidate gets all that state's electors. OK, the grand total number of electors is given to be 1001. Here is the difficulty I have with that: Both in the example with the nonconforming vector pop and erroneous computation with only 1000 electors being assigned, and in the vector pop in the puzzle itself, the number of electors is greater than the number of eligible voters! Last fiddled with by Dr Sardonicus on 20201202 at 00:41 Reason: xignif posty 
20201202, 02:44  #11 
Romulan Interpreter
Jun 2011
Thailand
3×47×67 Posts 

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