## Tribonacci sequence

Posted in Books, Kids, R with tags , , , , , on January 3, 2023 by xi'an

A simplistic puzzle from The Riddler when applying brute force:

A Tribonacci sequence is based on three entry integers a ≤ b ≤ c, and subsequent terms are the sum of the previous three. Among Tribonacci sequences containing 2023, which one achieves the smallest fourth term, a+b+c ?

The R code

tri<-function(a,b,e){
while(F<2023){
F=a+b+e;a=b;b=e;e=F}
return(F<2024)}
sol=NULL;m=674
for(a in 1:m)
for(b in a:m)
for(e in b:m)
if(tri(a,b,e)){
sol=rbind(sol,c(a,b,e))}


leads to (1,1,6) as the solution… Incidentally, this short exercise led me to finally look for a fix to entering vectors as arguments of functions requesting lists:

do.call("tri",as.list(sol[2023,]))


## Goats do room

Posted in Books, Kids, R, Statistics, Wines with tags , , , , , , , , , , , , on July 16, 2022 by xi'an

The riddle of the week is about 10 goats sequentially moving to their room, which they have chosen at random and independently (among ten rooms), unless another goat already occupies the room, in which case they move to the first free room with a higher number or fail. What is the probability that all goats end up in a room?

Coding the experiment is straightforward:

g=sample(1:N,N,rep=TRUE)
o=0*g
for(i in 1:N){
if(min(o[g[i]:N])){f=f+1;break()
}else{
o[min(which(!o[g[i]:N]))+g[i]-1]=1
}}}


returning an estimated probability of approximately 0.764.

As I had some free time during the early mornings at ISBA 2022, I tried to reformulate the question as a continuous event on uniform order statistics, turning to be at most one uniform larger than (N-1)/N, at most two larger than (N-2)/N, and so on… Asking the question on math.stackexchange quickly produced an answer that reversed engineered my formulation back to the goats (or parking lot), with a generic probability of

$\dfrac{(N+1)^{N-1}}{N^N}$

which of course coincides with the Monte Carlo approximation!

As an aside, I once drank South-African wines named Goats-do-Roam and Goat-Roti at my friends Jim and Maria’s place,  and they were quite enjoyable!

## self-avoiding random angles

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , on June 17, 2022 by xi'an

An apparently easy riddle from The Riddler this week: given N random half-lines starting from a N-S segment, what is the probability that none ever intersect? If m ½-lines are on the same side of the segment, they will not intersect when their angle with the segment is decreasing with the longitude of the endpoint of this ½-line on the segment. Assuming that drawing a ½-line at random is akin to uniformely drawing an angle on (0,π), this no X’ing event happens when the m angles are properly ordered, a 1/m! event. Independently, the probability for the segments on the other side is 1/(N-m)! The joint probability is thus

$\displaystyle\sum_{m=0}^N {N\choose m}\frac{1}{2^N} \frac{1}{m!(N-m)!}=\frac{(2N)!}{2^N(N!)^3}$

## top of the top

Posted in Statistics with tags , , , on August 19, 2021 by xi'an

An easy-peasy riddle from The Riddler about the probability that a random variable is the largest among ten iid variates, conditional on the event that this random variable is larger than the upper decile. This writes down easily as

$10\int_{q_{90}}^\infty F^9(x) f(x)\,\text d x$

if F and f are the cdf and pmf, respectively, which is equal to 1-.9¹⁰, approximately 1-e⁻¹, no matter what F is….

## more breaking sticks

Posted in Statistics with tags , , on July 19, 2021 by xi'an

A quick riddle from The Riddler that, when thinned to the actual maths problem, ends up asking for

$\mathbb P(\max(U_1,U_2,U_3)=U_3)+\mathbb P(\min(U_1,U_2,U_3)=U_3)$

which is equal to 2/3…. Anticlimactic.