Angkor finale [jatp]

and it only gets worse…

“You know, the saddest thing is that because I’m the president of the United States, I am not supposed to be involved with the Justice Department,” Mr. Trump said in a radio interview on Thursday on the “Larry O’Connor Show.” “I am not supposed to be involved with the F.B.I. I’m not supposed to be doing the kind of things that I would love to be doing. And I’m very frustrated by it.” NYT, Nov 03, 2017

“Two former US intelligence chiefs have said Donald Trump poses “a peril” to the US because he is vulnerable to being “played” by Russia, after the president said on Saturday he believed Vladimir Putin’s denials of Russian interference in the 2016 election.” The Guardian, Nov 12, 2017

“As a result [of the 44% of vacant seats in the appeal courts], Mr. Trump is poised to bring the conservative legal movement, which took shape in the 1980s in reaction to decades of liberal rulings on issues like the rights of criminal suspects and of women who want abortions, to a new peak of influence over American law and society.” NYT, Nov 11, 2017

“Hunting interests have scored a major victory with the Trump administration’s decision to allow Americans to bring home body parts of elephants shot for sport in Africa. Another totemic species now looks set to follow suit – lions.”  The Guardian, Nov 16, 2017

“Like everything else Trump touches, he hijacks it with his chronic dishonesty and childishness,” said Mark Salter, a longtime adviser to Senator John McCain, Republican of Arizona. “The intense, angry and largely ignorant tribalism afflicting our politics predates Trump’s arrival on the scene. But he has infused it with a psychopath’s inability to accept that social norms apply to him.” NYT, November  18, 2017

“We represent a much larger number of concerned mental health professionals who have come forward to warn against the president’s psychological instability and the dangers it poses. We now number in the thousands.” NYT, November 31, 2017

Le Monde puzzle [#872]

An “mildly interesting” Le Monde mathematical puzzle that eventually had me running R code on a cluster:

Within the set {1,…,56}, take 12 values at random, x₁,…,x₁₂. Is it always possible to pick two pairs from those 12 balls such that their sums are equal?

Indeed, while exhaustive search cannot reach the size of the set,

fowler=function(i=1,pred=NULL){
  pred=c(pred,i)
  for (j in (1:N)[-pred]){
     a=outer(c(pred,j),c(pred,j),"+")
     if ((sum(duplicated(a[lower.tri(a)]))>0)){
        val=FALSE
        }else{
          if (length(pred)==n-1){
            print(c(pred,j))
            val=TRUE
            }else{
            val=fowler(j,pred)}}
     if (val) break()
     }
  return(val)
  }
fowler(i=N,pred=1)

with N=35 being my upper limit (and n=9 the largest value inducing double sums), the (second) easiest verification goes by sampling as indicated and checking for duplicates.

mindup=66
for (t in 1:10^7){
#arguing that extremes should be included
  x=c(1,56,sample(2:55,10))
  A=outer(x,x,"+")
  mindup=min(mindup,sum(duplicated(A[lower.tri(A)])))
  if (mindup==0) break()}

The values of mindup obtained by running this code a few times are around 5, which means a certain likelihood of a positive answer to the above question…

This problem raises a much more interesting question, namely how to force simulations of those 12-uplets towards the most extreme values of the target function, from simulated annealing to cross-entropy to you-name-it… Here is my simulated annealing attempt:

target=function(x){
  a=outer(x,x,"+")
  return(sum(duplicated(a[lower.tri(a)])))}
beta=100
Nmo=N-1
nmt=n-2
nmo=n-1
x=sort(sample(2:Nmo,nmt))
cur=c(1,x,N)
tarcur=target(cur)
for (t in 1:10^6){
  dex=sample(2:nmo,2)
  prop=sort(c(cur[-dex],sample((2:Nmo)[-(cur-1)],2)))
  tarprop=target(prop)
  if (beta*log(runif(1))<tarprop -tarcur){
     cur=prop;tarcur=tarprop}
  beta=beta*.9999
  if (tarcur==0) break()}

Apart from this integer programming exercise, a few items of relevance in this Le Monde Science & Medicine leaflet.  A portrait of Leslie Lamport for his Turing Prize (yes, the very same Leslie Lamport who created LaTeX!, and wrote this book which stood on most mathematicians’ bookshelves for decades, with the marginally annoying lion comics at the head of each chapter!). A tribune on an interesting book, The Beginning and the End, by Clément Vidal, discussing how to prepare for the end of the Universe by creating a collective mind. And the rise of biobanks…