## Le Monde puzzle [open problem]

Posted in Books, Kids with tags , , , , , on October 23, 2017 by xi'an

What should have been the last puzzle in Le Monde competition turned out to be an anticlimactic fizzle on how many yes-no questions are needed to identify an integer between 1 and 1025=2¹⁰+1 and an extension to replies possibly being lies

What is much more exciting is that voting puzzle #1021 got cancelled because the authors of this puzzle thought the cascading majority rule would produce the optimal solution and it does not! (As exhibited by my R code.) So here is an open problem to ponder about! (And another puzzle in the pipeline to complete the competition.)

## Le Monde puzzle [#1024]

Posted in Books, Kids with tags , , , , , , , on October 10, 2017 by xi'an

The penultimate and appropriately somewhat Monty Hallesque Le Monde mathematical puzzle of the competition!

A dresser with 5×5 drawers contains a single object in one of the 25 drawers. A player opens a drawer at random and, after each choice, the object moves at random to a drawer adjacent to its current location and the drawer chosen by the player remains open. What is the maximum number of drawers one need to open to find the object?

In a dresser with 9 drawers in a line, containing again a single object, the player opens drawers one at a time, after which the open drawer is closed and the object moves to one of the drawers adjacent to its current location. What is the maximum number of drawers one need to open to find the object?

For the first question, setting a pattern of exploration and, given this pattern, simulating a random walk trying to avoid the said pattern as long as possible is feasible, returning a maximum number of steps over many random walks [and hence a lower bound on the true maximum]. As in the following code

sefavyd=function(pater=seq(1,49,2)%%25+1){
fild=matrix(0,5,5)
m=pater[1];i=fild[m]=1
t=sample((1:25)[-m],1)
nomove=FALSE
while (!nomove){
i=i+1
m=pater[i];fild[m]=1
if (t==m){ nomove=TRUE}else{
muv=NULL
if ((t-1)%%5>0) muv=c(muv,t-1)
if (t%%5>0) muv=c(muv,t+1)
if ((t-1)%/%5>0) muv=c(muv,t-5)
if (t%/%5<4) muv=c(muv,t+5)
muv=muv[fild[muv]==0]
nomove=(length(muv)==0)
if (!nomove) t=sample(rep(muv,2),1)}
}
return(i)}

But a direct reasoning starts from the observation that, while two adjacent drawers are not opened, a random walk can, with non-zero probability, switch indefinitely between both drawers. Hence, a sure recovery of the object requires opening one drawer out of two. The minimal number of drawers to open on a 5×5 dresser is 2+3+2+3+2=12. Since in 12 steps, those drawers are all open, spotting the object may require up to 13 steps.

For the second case, unless I [again!] misread the question, whatever pattern one picks for the exploration, there is always a non-zero probability to avoid discovery after an arbitrary number of steps. The [wrong!] answer is thus infinity. To cross-check this reasoning, I wrote the following R code that mimics a random pattern of exploration, associated by an opportunistic random walk that avoids discovery whenever possible (even with very low probability) bu pushing the object towards the centre,

drawl=function(){
i=1;t=5;nomove=FALSE
m=sample((1:9)[-t],1)
while (!nomove){
nextm=sample((1:9),1)
muv=c(t-1,t+1)
muv=muv[(muv>0)&(muv<10)&(muv!=nextm)]
nomove=(length(muv)==0)||(i>1e6)
if (!nomove) t=sample(rep(muv,2),1,
prob=1/(5.5-rep(muv,2))^4)
i=i+1}
return(i)}

which returns unlimited values on repeated runs. However, I was wrong and the R code unable to dismiss my a priori!, as later discussions with Robin and Julien at Paris-Dauphine exhibited ways of terminating the random walk in 18, then 15, then 14 steps! The idea was to push the target to one of the endpoints because it would then have no option but turning back: an opening pattern like 2, 3, 4, 5, 6, 7, 8, 8 would take care of a hidden object starting in an even drawer, while the following 7, 6, 5, 4, 3, 2 openings would terminate any random path starting from an odd drawer. To double check:

grawl=function(){
len=0;muvz=c(3:8,8:1)
for (t in 1:9){
i=1;m=muvz[i];nomove=(t==m)
while (!nomove){
i=i+1;m=muvz[i];muv=c(t-1,t+1)
muv=muv[(muv>0)&(muv<10)&(muv!=m)]
nomove=(length(muv)==0)
if (!nomove)
t=sample(rep(muv,2),1)}
len=max(len,i)}
return(len)}

produces the value 14.

## Le Monde puzzle [#1022 & #1023]

Posted in Books, Kids with tags , , , , , on September 29, 2017 by xi'an

Another Le Monde mathematical puzzle where I could not find a solution by R programming (albeit one by cissors and papers was readily available!):

An NT is a T whose head () is made of 3 50×50 squares and whose body (|) is made of N 50×50 squares.  What is the smallest possible side of a square containing four non-intersecting NT’s when N=1,2,4? And what is the smallest value of N such that this square also contains a fifth NT?

The questions could have been solved by brute force simulation (or a knapsack algorithm?!) but I could not fathom an efficient way to code throwing T’s at random over an MxM grid.So instead I took scissors and paper and tried to fit four 1T, 2T, and 4T into the smallest squares, ending up with 4×4, 5×5, and 7×7 squares. Interestingly, four 5T also fit in a 7×7 square. And a 9×9 square accommodates the extra 7T. Compared with the  “impossible” puzzle of last week, this is pretty anticlimactic..! (Actually, once the solutions were published, I realised the square containing the T’s did not have to be with integer side. Which means the smallest square for 3Ts was incorporating the glued T’s sideway. Fortunately, this did not impact the answer for the 7T’s!)

Going back to this “impossible” puzzle, the posted solution is somewhat… puzzling in that the resolution posits that the majority rule is the optimal allocation, when I am not sure it is [optimal]. Just because, when rerunning the same R code, I found instances when the minimal acceptable number of councillors was lower than the one returned by the majority rule.

And since this post get pushed down in the queue, here is as a bonus the equally anticlimactic puzzle #1023,

Find (a) a multiplication of two three-prime-digit numbers such that all digits everywhere in the long multiplication are prime and all three intermediary products have four prime digits, while the final result has six prime digits, and (b) a multiplication of two three-digit numbers such that the digits of the first one are odd (o), the digits of the second are even (e), the three intermediary products are all of the form eoe, and the final product is of the form eoeo. [The website has two pictures to help if this description is too unclear!]

This is indeed straightforward to code with one solution to (a) and two to (b) since the number of cases to examine is quite limited.

## weapons of math destruction [fan]

Posted in Statistics with tags , , , , , , , , on September 20, 2017 by xi'an

As a [new] member of Parliement, Cédric Villani is now in charge of a committee on artificial intelligence, which goal is to assess the positive and negative sides of AI. And refers in Le Monde interview below to Weapons of Maths Destruction as impacting his views on the topic! Let us hope Superintelligence is no next on his reading list…

## Le Monde puzzle [#1021]

Posted in Books, Kids, R with tags , , , , , on September 17, 2017 by xi'an

A puzzling Le Monde mathematical puzzle for which I could find no answer in the allotted time!:

A most democratic electoral system allows every voter to have at least one representative by having each of the N voters picking exactly m candidates among the M running candidates and setting the size n of the representative council towards this goal, prior to the votes. If there are M=25 candidates, m=10 choices made by the voters, and n=10 representatives, what is the maximal possible value of N? And if N=55,555 and M=33, what is the minimum value of n for which m=n is always possible?

I tried a brute force approach by simulating votes from N voters at random and attempting to find the minimal number of councillors for this vote, which only provides an upper bound of the minimum [for one vote], and a lower bound in the end [over all votes]. Something like

for (i in 1:N) votz[i,]=sample(1:M,n)
#exploration by majority
remz=1:N;conz=NULL
while (length(remz)>0){
seatz=order(-hist(votz[remz,],
breaks=(0:M)+0.5,plot=FALSE)\$density)[1]
conz=c(conz,seatz);nuremz=NULL
for (v in remz)
if (!(seatz%in%votz[v,])) nuremz=c(nuremz,v)
remz=nuremz}
solz=length(conz)
#exploration at random
kandz=matrix(0,N,M)
for (i in 1:N) kandz[i,votz[i,]]=1
for (t in 1:1e3){
#random choice of councillors
zz=sample(c(0,1),M,rep=TRUE)
while (min(kandz%*%zz)!=1)
zz=sample(c(0,1),M,rep=TRUE)
solz=min(solz,sum(zz))
#random choice of remaining councillor per voter
remz=1:N;conz=NULL
while (length(remz)>0){
seatz=sample(votz[remz[1],],1)
conz=c(conz,seatz);nuremz=NULL
for (i in remz)
if (!(seatz%in%votz[i,])) nuremz=c(nuremz,i)
remz=nuremz}
solz=min(solz,length(conz))}
maxz=max(solz,maxz)}

which leads to a value near N=4050 for the first question, with 0% confidence… Obviously, the problem can be rephrased as a binary integer linear programming problem of the form

$n= \max_A \min_{c;\,\min Ac=1}\mathbf{1}^\text{T}c$

where A is the NxM matrix of votes and c is the vector of selected councillors. But I do not see a quick way to fix it!

## Le Monde puzzle [#1020]

Posted in Books, Kids, R with tags , , , on September 15, 2017 by xi'an

A collection of liars in this Le Monde mathematical puzzle:

1. A circle of 16 liars and truth-tellers is such that everyone states that their immediate neighbours are both liars. How many liars can there be?
2. A circle of 12 liars and truth-tellers is such that everyone state that their immediate neighbours are one liar plus one truth-teller. How many liars can there be?
3.  A circle of 8 liars and truth-tellers is such that four state that their immediate neighbours are one liar plus one truth-teller and four state that their immediate neighbours are both liars . How many liars can there be?

These questions can easily be solved by brute force simulation. For the first setting, using 1 to code truth-tellers and -1 liars, I simulate acceptable configurations as

tabz=rep(0,16)
tabz[1]=1 #at least one
tabz[2]=tabz[16]=-1
for (i in 3:15){
if (tabz[i-1]==1){
tabz[i]=-1}else{
if (tabz[i+1]==-1){
tabz[i]=1}else{
if (tabz[i+1]==1){
tabz[i]=-1}else{
if (tabz[i-2]==-1){
tabz[i]=1}else{
tabz[i]=sample(c(-1,1),1)
}}}}}

which produces 8, 9, and 10 as possible (and obvious) values.

The second puzzle is associated with the similar R code

tabz=sample(c(-1,1),12,rep=TRUE)
rong=FALSE
while (!rong){
for (i in sample(12)){
if (tabz[i-1+12*(i==1)]*tabz[i%%12+1]==-1){
tabz[i]=1}else{
tabz[i]=sample(c(-1,1),1)}
}
rong=TRUE
for (i in (1:12)[tabz==1])
rong=rong&(tabz[i-1+12*(i==1)]*tabz[i%%12+1]==-1)
if (rong){
for (i in (1:12)[tabz==-1])
rong=rong&(tabz[i-1+12*(i==1)]*tabz[i%%12+1]!=-1)
}}

with numbers of liars (-1) either 12 (obvious) or 4.

The final puzzle is more puzzling in that figuring out the validating function (is an allocation correct?) took me a while, the ride back home plus some. I ended up with the following code that samples liars (-1) and thruth-seekers (1) at random, plus forces wrong and right answers (in 0,1,2) on these, and check for the number of answers of both types:

rong=FALSE
while (!rong){
tabz=sample(c(-1,1),8,rep=TRUE) #truth
tabz[1]=1;tabz[sample(2:8,1)]=-1
tt=(1:8)[tabz==1];lr=(1:8)[tabz==-1]
statz=rep(0,8) #stmt
statz[tt]=(tabz[tt-1+8*(tt==1)]*tabz[tt%%8+1]==-1)+
2*(tabz[tt-1+8*(tt==1)]+tabz[tt%%8+1]==-2)
statz[lr]=2*(tabz[lr-1+8*(lr==1)]*tabz[lr%%8+1]==-1)+
(tabz[lr-1+8*(lr==1)]+tabz[lr%%8+1]==-1)+
sample(c(1,2),8,rep=TRUE)[lr]*
(tabz[lr-1+8*(lr==1)]+tabz[lr%%8+1]==1)
rong=(sum(statz==1)==4)&(sum(statz==2)==4)}

with solutions 3, 4, 5 and 6.

## Texan black swan

Posted in Books, pictures with tags , , , , , , , , on September 12, 2017 by xi'an

“Un événement improbable aux conséquences d’autant plus désastreuses que l’on ne s’y est pas préparé.”

This weekend, there was a short article in Le Monde about the Harvey storm as a Texan illustration of Taleb’s black swan. An analysis that would imply every extreme event like this “once-in-a-thousand year” event (?) can be called a black swan… “An improbable event with catastrophic consequences, the more because it had not been provisioned”, as the above quote translates. Ironically, there is another article in the same journal, about the catastrophe being “ordinary” and “not unexpected”! While such massive floods are indeed impacting a huge number of people and companies, because the storm happened to pour an unusual amount of rain right on top of Houston, they indeed remain within the predictable and not so improbable in terms of the amount of water deposited in the area and in terms of damages, given the amount and style of construction over flood plains. For instance, Houston is less than 50 feet above sea level, has fairly old drainage and pipe systems, and lacks a zoning code. With mostly one or two-story high buildings rather than higher rises. (Incidentally, I appreciated the juxtaposition of the article with the add for Le Monde des Religions and its picture of a devilesque black goat!)