## easy Riddler

Posted in Kids, R with tags , , , on May 10, 2019 by xi'an

The riddle of the week is rather standard probability calculus

If N points are generated at random places on the perimeter of a circle, what is the probability that you can pick a diameter such that all of those points are on only one side of the newly halved circle?

Since it is equivalent to finding the range of N Uniform variates less than ½. And since the range of N Uniform variates is distributed as a Be(N-1,2) random variate. The resulting probability, which happens to be exactly $N/2^{N-1}$, is decreasing exponentially, as shown below…

## survivalists [a Riddler’s riddle]

Posted in Books, Kids, R, Statistics with tags , , , , , , on April 22, 2019 by xi'an

A neat question from The Riddler on a multi-probability survival rate:

Nine processes are running in a loop with fixed survivals rates .99,….,.91. What is the probability that the first process is the last one to die? Same question with probabilities .91,…,.99 and the probability that the last process is the last one to die.

The first question means that the realisation of a Geometric G(.99) has to be strictly larger than the largest of eight Geometric G(.98),…,G(.91). Given that the cdf of a Geometric G(a) is [when counting the number of attempts till failure, included, i.e. the Geometric with support the positive integers]

$F(x)=\Bbb P(X\le x)=1-a^{x}$

the probability that this happens has the nice (?!) representation

$\sum_{x=2}^\infty a_1^{x-1}(1-a_1)\prod_{j\ge 2}(1-a_j^{x-1})=(1-a_1)G(a_1,\ldots,a_9)$

which leads to an easy resolution by recursion since

$G(a_1,\ldots,a_9)=G(a_1,\ldots,a_8)-G(a_1a_9,\ldots,a_8)$

and $G(a)=a/(1-a)$

and a value of 0.5207 returned by R (Monte Carlo evaluation of 0.5207 based on 10⁷ replications). The second question is quite similar, with solution

$\sum_{x=2}^\infty a_1^{x-1}(1-a_1)\prod_{j\ge 1}(1-a_j^{x})=a^{-1}(1-a_1)G(a_1,\ldots,a_9)$

and value 0.52596 (Monte Carlo evaluation of 0.52581 based on 10⁷ replications).

## numbers of numbers with numbers of their numbers

Posted in Statistics with tags , , , on April 6, 2019 by xi'an

A funny riddle from The Riddler where the number of numbers that indicate the numbers of their numbers is requested. By which one means numbers like 22 (two 2’s) and 21322314 (two 1’s, three 2’s, two 3’s and one 4), the convention being that “numbers consist of alternating tallies and numerals”. And that numerals are “tallied in increasing order”. A reasoning based on the number of 1’s, from zero (where 22 seems to be the only possibility) to six (where 613223141516171819 is the only case) leads to a total of 59 cases (unless zero counts as an extra case) and a brute force R exploration returns the same figure:

for (t in 1:1e5){
az=sample((0:6),9,rep=TRUE)
count=rep(TRUE,9)
for (i in 1:9) count[i]=(sum(az==i)+(az[i]>0))==az[i]
nit=0
while ((min(count)==0)&(nit<1e2)){
j=sample((1:9)[!count],1)
az[j]=(sum(az==j)+(az[j]>0))
nit=nit+1}
if (min(count)==1) solz=unique.matrix(rbind(solz,az),mar=1)}


The solution published on The Riddler differs because it also includes numbers with zeros. Which I find annoying to the extreme because if zeroes are allowed then every digit not in the original solution should appear as multiplied by 0, which is self-contradictory… For instance 22 should be 012203…09, except then there is one 1, one 3, and so on.

## no country for old liars

Posted in Kids, R with tags , , , , , on March 30, 2019 by xi'an

A puzzle from the Riddler about a group of five persons, A,..,E, where all and only people strictly older than L are liars, all making statements about others’ ages:

1. A: B>20 and D>16
2. B: C>18 and E<20
3. C: D<22 and A=19
4. D: E≠20 and B=20
5. E: A>21 and C<18

The Riddler is asking for the (integer value of L and the ranges or values of A,…,E. After thinking about this puzzle over a swimming session, I coded the (honest) constraints and their (liar) complements as many binary matrices, limiting the number of values of L to 8 from 0 (15) to 7 (22) and A,…,E to 7 from 1 (16) to 7 (22):

CA=CB=CC=CD=CE=A=B=C=D=E=matrix(1,5,7)
#constraints
A[2,1:(20-15)]=A[4,1]=0 #A honest
CA[2,(21-15):7]=CA[4,2:7]=0 #A lying
B[3,1:(18-15)]=B[5,(20-15):7]=0
CB[3,(19-15):7]=CB[5,1:(19-15)]=0
C[1,-(19-15)]=C[4,7]=0 #C honest
CC[1,(19-15)]=CC[4,-7]=0 #C lying
D[5,(17-15)]=D[2,-(20-15)]=0
CD[5,-(17-15)]=CD[2,(20-15)]=0
E[1,1:(21-15)]=E[3,(18-15):7]=0
CE[1,7]=CE[3,1:(17-15)]=0


since the term-wise product of these five matrices expresses all the constraints on the years, as e.g.

ABCDE=A*CB*CC*D*CE


if A,D≤L and B,C,E>L, and I then looked by uniform draws [with a slight Gibbs flavour] for values of the integers that suited the constraints or their complement, the stopping rule being that the collection of A,…,E,L is producing an ABCDE binary matrix that agrees with all statements modulo the lying statuum of their authors:

yar=1:5
for (i in 1:5) yar[i]=sample(1:7,1)
L=sample(0:7,1)
ABCDE=((yar[1]>L)*CA+(yar[1]<=L)*A)*
((yar[2]>L)*CB+(yar[2]<=L)*B)*
((yar[3]>L)*CC+(yar[3]<=L)*C)*
((yar[4]>L)*CD+(yar[4]<=L)*D)*
((yar[5]>L)*CE+(yar[5]<=L)*E)
while (min(diag(ABCDE[,yar]))==0){
L=sample(0:7,1);idx=sample(1:5,1)
if (max(ABCDE[idx,])==1) yar[idx]=sample(which(ABCDE[idx,]>0),1)
ABCDE=((yar[1]>L)*CA+(yar[1]<=L)*A)*
((yar[2]>L)*CB+(yar[2]<=L)*B)*
((yar[3]>L)*CC+(yar[3]<=L)*C)*
((yar[4]>L)*CD+(yar[4]<=L)*D)*
((yar[5]>L)*CE+(yar[5]<=L)*E)
}


which always produces L=18,A=19,B=20,C=18,D=16 and E>19 as the unique solution (also reported by The Riddler).

> ABCDE
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    0    0    1    0    0    0
[2,]    0    0    0    0    1    0    0
[3,]    0    0    1    0    0    0    0
[4,]    1    0    0    0    0    0    0
[5,]    0    0    0    0    1    1    1


## take a random integer

Posted in Books, Statistics with tags , , on February 16, 2019 by xi'an

A weird puzzle from FiveThirtyEight: what is the probability that the product of three random integers is a multiple of 100? Ehrrrr…, what is a random integer?! The solution provided by the Riddler is quite stunning

Reading the question charitably (since “random integer” has no specific meaning), there will be an answer if there is a limit for a uniform distribution of positive integers up to some number . But we can ignore that technicality, and make do with the idealization that since every second, fourth, fifth, and twenty-fifth integer are divisible by and , the chances of getting a random integer divisible by those numbers are , , , and .

as it acknowledges that the question is meaningless, then dismisses this as a “technicality” and still handles a Uniform random integer on {1,2,…,N} as N grows to infinity! Since all that matters is the remainder of the “random variable” modulo 100, this remainder will see its distribution vary as N moves to infinity, even though it indeed stabilises for $N$ large enough…

## missing digit in a 114 digit number [a Riddler’s riddle]

Posted in R, Running, Statistics with tags , , , , , , , on January 31, 2019 by xi'an

A puzzling riddle from The Riddler (as Le Monde had a painful geometry riddle this week): this number with 114 digits

530,131,801,762,787,739,802,889,792,754,109,70?,139,358,547,710,066,257,652,050,346,294,484,433,323,974,747,960,297,803,292,989,236,183,040,000,000,000

is missing one digit and is a product of some of the integers between 2 and 99. By comparison, 76! and 77! have 112 and 114 digits, respectively. While 99! has 156 digits. Using WolframAlpha on-line prime factor decomposition code, I found that only 6 is a possible solution, as any other integer between 0 and 9 included a large prime number in its prime decomposition:

However, I thought anew about it when swimming the next early morning [my current substitute to morning runs] and reasoned that it was not necessary to call a formal calculator as it is reasonably easy to check that this humongous number has to be divisible by 9=3×3 (for else there are not enough terms left to reach 114 digits, checked by lfactorial()… More precisely, 3³³x33! has 53 digits and 99!/3³³x33! 104 digits, less than 114), which means the sum of all digits is divisible by 9, which leads to 6 as the unique solution.

## running plot [and simulated annealing]

Posted in Kids, pictures, R with tags , , on December 14, 2018 by xi'an

Last weekend, I found out a way to run updated plots within a loop in R, when calling plot() within the loop was never updated in real time. The above suggestion of including a Sys.sleep(0.25) worked perfectly on a simulated annealing example for determining the most dispersed points in a unit disc.