December 2020

tgan · 2020-12-01, 07:19

December 2020.

tgan · 2020-12-01, 07:41

134 160 206 235 265 = 1000 and not 1001

I think the correct numnber should be 134 161 206 235 265

tgan · 2020-12-01, 09:27

Quote:

Originally Posted by tgan

134 160 206 235 265 = 1000 and not 1001

I think the correct numnber should be 134 161 206 235 265

And [50, 60, 77, 88, 99] only 99 is odd

LaurV · 2020-12-01, 09:58

Yep, you are right on both counts. They will probably correct it later.
I fixed your first post (runaway link).

retina · 2020-12-01, 12:33

I count 67 unique solutions.

Have puzzles in the past had multiple solutions, or did I just make a mistake somewhere?

tgan · 2020-12-01, 14:56

Quote:

Originally Posted by retina

I count 67 unique solutions.

Have puzzles in the past had multiple solutions, or did I just make a mistake somewhere?

I think it is possible
must admit that i did not totally understood the puzzle. do we look for a maximum or we need to get the required number?

EdH · 2020-12-01, 15:24

Quote:

Originally Posted by retina

I count 67 unique solutions.

Have puzzles in the past had multiple solutions, or did I just make a mistake somewhere?

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.

Dr Sardonicus · 2020-12-01, 15:44

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 non-negative even integers.

Dividing through by 2, we have the the problem of expressing 498 as the sum of 5 non-negative integers.

This is well-known to be equal to the number of ways of expressing 498 as the sum

498 = x₁ + 2*x₂ + 3*x₃ + 4*x₄ + 5*x₅

where the x's are non-negative integers; the corresponding 5 summands of 498 are

x₅, x₅ + x₄, x₅ + x₄ + x₃, x₅ + x₄ + x₃ + x₂, and x₅ + x₄ + x₃ + x₂ + x₁.

The number of such 5-tuples is approximately the volume of the 5-dimensional "simplex" in Euclidean 5-space 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.

retina · 2020-12-01, 22:28

Quote:

Originally Posted by Dr Sardonicus

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.

If you ignore the ordering, and just consider the raw population counts, then 67 results. If you account for the ordering then a lot more than 67 results.

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 non-conforming examples.

Dr Sardonicus · 2020-12-02, 00:39

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 non-conforming 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!

LaurV · 2020-12-02, 02:44

Quote:

Originally Posted by retina

It all seems a bit muddled

Of course, what did you expect? It is about elections...

2020-12-01, 07:19	#1
tgan Jul 2015 2³×5 Posts	December 2020 December 2020. Last fiddled with by LaurV on 2020-12-01 at 07:21 Reason: fixed link

2020-12-01, 07:41	#2
tgan Jul 2015 101000₂ 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 2020-12-01 at 07:43

2020-12-01, 15:44	#8
Dr Sardonicus Feb 2017 Nowhere 2⁶·3²·11 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 non-negative even integers. Dividing through by 2, we have the the problem of expressing 498 as the sum of 5 non-negative integers. This is well-known to be equal to the number of ways of expressing 498 as the sum 498 = x₁ + 2x₂ + 3x₃ + 4x₄ + 5x₅ where the x's are non-negative integers; the corresponding 5 summands of 498 are x₅, x₅ + x₄, x₅ + x₄ + x₃, x₅ + x₄ + x₃ + x₂, and x₅ + x₄ + x₃ + x₂ + x₁. The number of such 5-tuples is approximately the volume of the 5-dimensional "simplex" in Euclidean 5-space 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 2020-12-01 at 16:14 Reason: Forgot extra factor of 5! in denominator*

2020-12-02, 00:39	#10
Dr Sardonicus Feb 2017 Nowhere 2⁶·3²·11 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 2v_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 non-conforming 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 2020-12-02 at 00:41 Reason: xignif posty

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2020-12-01, 09:58	#4
LaurV Romulan Interpreter "name field" Jun 2011 Thailand 5×11²×17 Posts	Yep, you are right on both counts. They will probably correct it later. I fixed your first post (runaway link).

2020-12-01, 12:33	#5
retina Undefined "The unspeakable one" Jun 2006 My evil lair 2⁶×3×5×7 Posts	I count 67 unique solutions. Have puzzles in the past had multiple solutions, or did I just make a mistake somewhere?