S*****H
发帖数: 90 | 1 1. Is it possible to change the numbers on two six sided dice to other
positive numbers so that the probability distribution of their sum remains
unchanged?
2. N points lie on a circle. You draw lines connecting all the points to
each other. These lines divide up the circle into a number of regions. How
many regions is this? Assume that the points are scattered in such a way as
to give the maximum number of regions for that N. | v********g
发帖数: 14 | 2 Q2:
N points will have N(N-1)/2 lines.
To add kth line, number of region is: NR(k)=NR(k-1)+k
Thus the number region for N points:
1+N(N-1)/4+N^2(N-1)^2/8.
Is that right? | t*******y
发帖数: 637 | 3 1.
1 2 3 4 5 6
1 7 13 19 25 31?
as
【在 S*****H 的大作中提到】
: 1. Is it possible to change the numbers on two six sided dice to other
: positive numbers so that the probability distribution of their sum remains
: unchanged?
: 2. N points lie on a circle. You draw lines connecting all the points to
: each other. These lines divide up the circle into a number of regions. How
: many regions is this? Assume that the points are scattered in such a way as
: to give the maximum number of regions for that N.
| s*******u
发帖数: 35 | 4 1 貌似不行
as
【在 S*****H 的大作中提到】
: 1. Is it possible to change the numbers on two six sided dice to other
: positive numbers so that the probability distribution of their sum remains
: unchanged?
: 2. N points lie on a circle. You draw lines connecting all the points to
: each other. These lines divide up the circle into a number of regions. How
: many regions is this? Assume that the points are scattered in such a way as
: to give the maximum number of regions for that N.
| p****i
发帖数: 83 | 5 a_(n 1) = ((n-6) a_n)/(n-7)-7/(n-7) (for all n>=8)
1.1 2 3 4 5 61 7 13 19 25 31?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Sent from iPhone App: Mitbbs 3.1 - iPad Paid
【在 S*****H 的大作中提到】
: 1. Is it possible to change the numbers on two six sided dice to other
: positive numbers so that the probability distribution of their sum remains
: unchanged?
: 2. N points lie on a circle. You draw lines connecting all the points to
: each other. These lines divide up the circle into a number of regions. How
: many regions is this? Assume that the points are scattered in such a way as
: to give the maximum number of regions for that N.
| k**x
发帖数: 2611 | 6 1. 两个筛子上的数字需要一样么？不需要一样的话，第一个每个数字减x，第二个每个
数字加x就行了。0
需要一样的的，从最小的开始往上推应该可以推出1,2,3,4,5,6是唯一解。
i.e, 最小的和是2，必然是1+1，然后第二小的和肯定就是1+2=3，依此类推。
as
【在 S*****H 的大作中提到】
: 1. Is it possible to change the numbers on two six sided dice to other
: positive numbers so that the probability distribution of their sum remains
: unchanged?
: 2. N points lie on a circle. You draw lines connecting all the points to
: each other. These lines divide up the circle into a number of regions. How
: many regions is this? Assume that the points are scattered in such a way as
: to give the maximum number of regions for that N.
| S*****H
发帖数: 90 | 7
两个筛子上的数字不需要一样，但是０不是正整数。
这个问题是很奇妙的。我第一印象是不存在，但是却无法证明。
我试了连续分布，理论上是可行的。所以又试图构造，居然找到了一个解答。
【在 k**x 的大作中提到】
: 1. 两个筛子上的数字需要一样么？不需要一样的话，第一个每个数字减x，第二个每个
: 数字加x就行了。0: 需要一样的的，从最小的开始往上推应该可以推出1,2,3,4,5,6是唯一解。
: i.e, 最小的和是2，必然是1+1，然后第二小的和肯定就是1+2=3，依此类推。
:
: as
| c********y
发帖数: 30813 | 8 可以证明的，有且只有另外一种可能。
【在 S*****H 的大作中提到】
:
: 两个筛子上的数字不需要一样，但是０不是正整数。
: 这个问题是很奇妙的。我第一印象是不存在，但是却无法证明。
: 我试了连续分布，理论上是可行的。所以又试图构造，居然找到了一个解答。
| S*****H
发帖数: 90 | 9 不见得吧。连续分布的话，有无穷种可能。
但是离散分布的话，就有可能不存在。你可以试一试骰子有三个面(1,2,3)的情形。
【在 c********y 的大作中提到】
: 可以证明的，有且只有另外一种可能。
| c********y
发帖数: 30813 | 10 说的就是6面的骰子。
除了standard dice外，只有一种可能，是可以证明的。而且证明很有意思的，我就不
剧透了。
【在 S*****H 的大作中提到】
: 不见得吧。连续分布的话，有无穷种可能。
: 但是离散分布的话，就有可能不存在。你可以试一试骰子有三个面(1,2,3)的情形。
| | | p*****k
发帖数: 318 | 11 SwingLH,
(1) [1,3,4,5,6,8] and [1,2,2,3,3,4] by considering
the prob generating function. see:
http://www.wilmott.com/messageview.cfm?catid=26&threadid=66564
(somehow cannot find the old thread here)
(2) see
http://www.research.att.com/~njas/sequences/A000127 | b*********r
发帖数: 302 | 12 One simple solution is: 3,3,3,3,3,3 and 4,4,4,4,4,4 | S*****H
发帖数: 90 | 13 pcasnik,
Thank you very much. You are a real guru.
【在 p*****k 的大作中提到】
: SwingLH,
: (1) [1,3,4,5,6,8] and [1,2,2,3,3,4] by considering
: the prob generating function. see:
: http://www.wilmott.com/messageview.cfm?catid=26&threadid=66564
: (somehow cannot find the old thread here)
: (2) see
: http://www.research.att.com/~njas/sequences/A000127
| S*****H
发帖数: 90 | 14 Thank you. I think that I got the rough idea. Basically prove that the
generating function (x+x^2+...+x^6)^2 has only one alternative way to be
written to product of two 6-term non-constant-term polynomials.
【在 c********y 的大作中提到】
: 说的就是6面的骰子。
: 除了standard dice外，只有一种可能，是可以证明的。而且证明很有意思的，我就不
: 剧透了。
| S*****H
发帖数: 90 | 15 Pleaset note that the sum distribution is
P(sum=2)=1/36,
P(sum=3)=2/36,
... ...
P(sum=7)=6/36,
... ...
P(sum=12)=1/36.
【在 b*********r 的大作中提到】
: One simple solution is: 3,3,3,3,3,3 and 4,4,4,4,4,4
| g******e
发帖数: 352 | 16 1. 筛子上的数必须是整数吗？
如果不需要是整数的话，
Dice 1: [0.5, 1.5, 2.5, 3.5, ... 5.5]
Dice 2: [1.5, 2.5, 3.5, 4.5, ... 6.5]
我是不是漏想了什么？？
as
【在 S*****H 的大作中提到】
: 1. Is it possible to change the numbers on two six sided dice to other
: positive numbers so that the probability distribution of their sum remains
: unchanged?
: 2. N points lie on a circle. You draw lines connecting all the points to
: each other. These lines divide up the circle into a number of regions. How
: many regions is this? Assume that the points are scattered in such a way as
: to give the maximum number of regions for that N.
| c**********e
发帖数: 2007 | 17 应该是整数吧。
【在 g******e 的大作中提到】
: 1. 筛子上的数必须是整数吗？
: 如果不需要是整数的话，
: Dice 1: [0.5, 1.5, 2.5, 3.5, ... 5.5]
: Dice 2: [1.5, 2.5, 3.5, 4.5, ... 6.5]
: 我是不是漏想了什么？？
:
: as
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