Archive for Paris

Le Monde puzzle [#1157]

Posted in Books, Kids, R with tags , , , , , , on October 1, 2020 by xi'an

The weekly puzzle from Le Monde is an empty (?) challenge:

Kimmernaq and Aputsiaq play a game where Kimmernaq picks ten different integers between 1 and 100, and Aputsiaq must find a partition of these integers into two groups with identical sums. Who is winning?

Indeed, if the sums are equal, then the sum of their sums is even, meaning the sum of the ten integers is even. Any choice of these integers such that the sum is odd is a sure win for Kimmernaq. End of the lame game (if I understood its wording correctly!). If some integers can be left out of the groups, then both solutions seem possible: calling the R code

P=1;M=1e3
while (P<M){
a=sample(1:M,10);P=Q=0
while((P<M)&(!Q)){
t=sample(1:7,1)     #size of subset
o=1         #total sum must be even
while(!!(s<-sum(o))%%2)o=sample(a,10-t)
Q=max(2*cumsum(b<-sample(o))==s)
P=P+1}}

I found no solution (i.e. exiting the outer while loop) for M not too large…  So Aputsiaq is apparently winning. Le Monde solution considers the 2¹⁰-1=1023 possible sums made out of 10 integers, which cannot exceed 955, hence some of these sums must be equal (and the same applies when removing the common terms from both sums!). When considering the second half of the question

What if Kimmernaq picks 6 distinct integers between 1 and 40, and Aputsiaq must find a partition of these integers into two groups with identical sums. Who is winning?

recycling the above R code produced subsets systematically hitting the upper bound M, for much larger values. So Kimmernaq should have a mean to pick 6 integers such that any subgroup cannot be broken into two parts with identical sums. One of the outcomes being

 
> a
[1] 36 38 30 18  1 22

one can check that all the possible sums differ:

aa=a
for(i in 2:5){
 bb=NULL
 while(length(bb)<choose(6,i))bb=unique(c(bb,sum(sample(a,i))))
 aa=c(aa,bb)}
unique(aa)

and the outcome is indeed of length 2⁶-2=62!

As an aside, a strange [to me at least] R “mistake” was that when recycling the variable F in a code-golfing spirit, since it is equal to zero by default, rather than defining a new Q:

while((P<M)&(!F)){
...
F=max(2*cumsum(b<-sample(o))==s)
P=P+1}

the counter P was not getting updated!

Le Monde puzzle [#1155]

Posted in Books, Kids, R with tags , , , , , , , on September 26, 2020 by xi'an

The weekly puzzle from Le Monde is another Sudoku challenge:

Anahera and Wiremu play a game for T rounds. They successively pick a digit between 1 and 3, never repeating the previous one, and sum these digits. The last to play wins if the sum is a multiple of 3. Who is the winner for an optimal strategy?

By a simple dynamic programming of the optimal strategy at each step

N=3
A=matrix(-1,20,N)
A[1,1:N]=1:N
for (T in 2:20)
for (i in 1:N) A[T,i]=i+ifelse(!T%%2, #parity check
max((i+A[T-1,-i])%%3), #avoid zero
min((i+A[T-1,-i])%%3)) #seek zero

the first to play can always win the game. Not fun!

c’est reparti ! [bis]

Posted in Books, Kids, pictures, Running with tags , , , , , , , on September 25, 2020 by xi'an

Creparti

homeless hosted in my former office

Posted in pictures, Travel with tags , , , , , , , , , on September 24, 2020 by xi'an

launch of ELLIS

Posted in Statistics, University life with tags , , , , , , , , , on September 15, 2020 by xi'an

The European Laboratory for Learning and Intelligent Systems (ELLIS) is a network (inspired by the Canadian LMB CIFAR network ) that has been recently created to keep European research in artificial intelligence and machine learning at the forefront, to keep up with North America and China where the AI investments are far superior. It has currently 30 units and will be officially launched this Tuesday, 15 September with live streaming. (I am part of the Paris Ellis Unit, directed by Gabiel Peyré.) It also organizes PhD and postdoc exchange programs.