Matlab goes deep [learning]

A most interesting link I got when reading Le Monde, about MatLab proposing deep learning tools…

art brut

Le Monde puzzle [#965]

A game-related Le Monde mathematical puzzle:

Starting with a pile of 10⁴ tokens, Bob plays the following game: at each round, he picks one of the existing piles with at least 3 tokens, takes away one of the tokens in this pile, and separates the remaining ones into two non-empty piles of arbitrary size. Bob stops when all piles have identical size. What is this size and what is the maximal number of piles?

First, Bob can easily reach a decomposition that prevents all piles to be of the same size: for instance, he can start with a pile of 1 and another pile of 2. Looking at the general perspective, an odd number of tokens, n=2k+1, can be partitioned into (1,1,2k-1). Which means that the decomposition (1,1,…,1) involving k+1 ones can always be achieved. For an even number, n=2k, this is not feasible. If the number 2k can be partitioned into equal numbers u, this means that the sequence 2k-(u+1),2k-2(u+1),… ends up with u, hence that there exist m such that 2k-m(u+1)=u or that 2k+1 is a multiple of (u+1). Therefore, the smallest value is made of the smallest factor of 2k+1. Minus one. For 2k=10⁴, this value is equal to 72, while it is 7 for 10³. The decomposition is impossible for 2k=100, since 101 is prime. Here are the R functions used to check this analysis (with small integers, if not 10⁴):

solvant <- function(piles){
 if ((length(piles)>1)&((max(piles)==2)||(min(piles)==max(piles)))){
  return(piles)}else{
   i=sample(rep(1:length(piles),2),1,prob=rep(piles-min(piles)+.1,2))
   while (piles[i]<3)
    i=sample(rep(1:length(piles),2),1,prob=rep(piles-min(piles)+.1,2))
   split=sample(rep(2:(piles[i]-1),2),1,
        prob=rep(abs(2:(piles[i]-1)-piles[i]/2)+.1,2))
   piles=c(piles[-i],split-1,piles[i]-split)
   solvant(piles)}}

disolvant <- function(piles){
 sol=solvant(piles)
 while (min(sol)<max(sol))
 sol=solvant(piles)
 return(sol)}

resolvant <- function(piles){
 sol=disolvant(piles)
 lasol=sol;maxle=length(sol)
 for (t in 1:piles){
 sol=disolvant(piles)
 if (length(sol)>maxle){
 lasol=sol;maxle=length(sol)}}
 return(lasol)}

Le Monde puzzle [#964]

A not so enticing Le Monde mathematical puzzle:

Find the minimal value of a five digit number divided by the sum of its digits.

This can formalised as finding the minimum of N/(a+b+c+d+e) when N writes abcde. And solved by brute force. Using a rough approach to finding the digits of a five-digit number, the question can be easily solved as

pris=1e6
for (i in 1e4:1e5){
 pres=i/sum((i %% 10^{5:1})) %/% 10^{4:0})
 if (pres<pris){
 pris=pres;sol=i}}

which returns N=10999 as its solution. (The solution for six digits is 10999.) The mathematical solution as provided in the newspaper certainly sounded more exciting.

Le Monde puzzle [#960]

An arithmetic Le Monde mathematical puzzle:

Given an integer k>1, consider the sequence defined by F(1)=1+1 mod k, F²(1)=F(1)+2 mod k, F³(1)=F²(1)+3 mod k, &tc. [With this notation, F is not necessarily a function.] For which value of k is the sequence the entire {0,1,…,k-1} set?

This leads to an easy brute force resolution, for instance writing the R function

crkl<-function(k) return(unique(cumsum(1:(2*k))%%k))

where 2k is a sufficient substitute for ∞. Then the cases where the successive images of 1 visit the entire set {0,1,…,k-1} are given by

> for (i in 2:550) if (length(crkl(i))==i) print(i)
[1] 2
[1] 4
[1] 8
[1] 16
[1] 32
[1] 64
[1] 128
[1] 256
[1] 512

which suspiciously looks like the result that only the powers of 2 k=2,2²,2³,… lead to a complete exploration of the set {0,1,…,k-1}. Checking a few series in the plane back from Warwick, I quickly found that when k is odd, (1) the sequence is of period k and (2) there is symmetry in the sequence, which means it only takes (k-1)/2 values. For k even, there is a more complicated symmetry, with the sequence being of period 2k, symmetric around its two middle values, and taking the values 1,2,..,1+k(2k+1)/4,..,1+k(k+1)/2. Those values cannot cover the set {0,1,…,k-1} if two are equal, which means an i(i+1)/2 congruent to zero modulo k, hence equal to k. This is clearly impossible when k is a power of 2 because i and i+1 cannot both divide a power of 2. I waited for the published solution as of yesterday’s and the complete argument was to show that when N=2p, the corresponding sequence [for N] is made (modulo p) of the sequence for p plus the same sequence translated by p. The one for N is complete only if the one for p is complete, which by recursion eliminates all cases but the powers of 2…

Le Monde puzzle [#959]

Another of those arithmetic Le Monde mathematical puzzle:

Find an integer A such that A is the sum of the squares of its four smallest dividers (including1) and an integer B such that B is the sum of the third poser of its four smallest factors. Are there such integers for higher powers?

This begs for a brute force resolution checking the integers until a solution appears. The only exciting part is providing the four smallest factors but a search on Stack overflow led to an existing R function:

FUN <- function(x) {
    x <- as.integer(x)
    div <- seq_len(abs(x))
    return(div[x %% div == 0L])
}

(which uses the 0L representation I was unaware of) and hence my R code:

quest1<-function(n=2){
 I=4
 stop=TRUE
 while ((stop)&(I<1e6)){
  I=I+1
  dive=FUN(I)
  if (length(dive)>3)
     stop=(I!=sum(sort(dive)[1:4]^n))
   }
 return(I)
 }

But this code only seems to work for n=2 as produces A=130: it does not return any solution for the next value of n… As shown by the picture below, which solely exhibits a solution for n=2,5, A=17864 (in the second case), there is no solution less than 10⁶ for n=3,4,6,..9. So, unless I missed a point in the question, the solutions for n>2 are larger if they at all exist.

A resolution got published yesterday night in Le Monde  and (i) there is indeed no solution for n=3 (!), (ii) there are solutions for n=4 (1,419,874) and n=5 (1,015,690), which are larger than the 10⁶ bound I used in the R code, (iii) there is supposedly no solution for n=5!, when the R code found that 17,864=1⁵+2⁵+4⁵+7⁵… It is far from the first time the solution is wrong or incomplete!

Le Monde puzzle [#958]

A knapsack Le Monde mathematical puzzle:

Given n packages weighting each at most 5.8kg for a total weight of 300kg, is it always possible to allocate these packages  to 12 separate boxes weighting at most 30kg each? weighting at most 29kg each?

This can be checked by brute force using the following R code

#generate packages
paca=runif(1,0,5.8)
while (sum(paca)<300){ 
  paca=c(paca,runif(1,0,5.8))} 
paca=paca[-length(paca)] 
paca=c(paca,300-sum(paca)) 

and

#check if they can be partitioned into 
#12 groups of weight less than 30 
box=vector(mode="list",length=12) 
#random allocation 
alloc=sample(1:12,length(paca),rep=TRUE) 
content=rep(0,12) 
for (i in 1:12){ 
box[[i]]=paca[alloc==i] 
content[i]=sum(box[[i]])} 
content=content*(content>0) 
#wrong allocation 
 while (max(content)>30){
  i=sample(1:12,1,prob=content)
  j=sample(1:length(box[[i]]),1,prob=box[[i]])
#reallocation
  k=sample(1:12,1,prob=max(content)-content)
  while (k==i){
    k=sample(1:12,1,prob=max(content)-content)}
  content[i]=content[i]-box[[i]][j]
  content[i]=content[i]*(content[i]>0)
  content[k]=content[k]+box[[i]][j]
  box[[k]]=c(box[[k]],box[[i]][j])
  box[[i]]=box[[i]][-j]}

repeatedly and could not find an endless while loop. (Empty boxes sometimes lead to negative contents, hence my fix setting negative contents to zero.) But neither did I find an issue when the upper bound on the weight is 29kg… So it is either possible or rarely impossible! The R code immediately gets stuck when setting the bound at 25kg.

After reading the solution of April 13 in Le Monde, it appears that there is a counter example for the 29kg limit, namely 60 packages weighting 4.91kg plus one package weighting 5.4kg, since the first 60 packages fit inside 12 boxes and the last one is left out.