## Le Monde puzzle [#1016]

Posted in Books, Kids with tags , , , on July 16, 2017 by xi'an

An even more straightforward Le Monde mathematical puzzle that took a few minutes to code in the train to Cambridge:

1. Breaking {1,…,8} into two sets of four integrals, what is (or are) the division into two groups of equal size such that the sums of the squared terms from each are equal? Same question for the set {21,…,28}.
2.  Considering the integers from 1 to 12, how many divisions into two groups of size six satisfy the above property? Same question when the two groups are of different sizes.

The first code is

nop=TRUE
while (nop){
s=sample(1:8)
nop=(sum(s[1:4]^2)!=sum(s[5:8]^2))}


with result

1 6 4 7

while the second set leads to the unique [drifted] solution (up to symmetries)

21 24 26 27


and the divisions for the larger set {1,…,12} is unique in the equal case, and are four in the unequal case.

## easy riddle

Posted in Books, Kids, R with tags , , , , , on July 12, 2017 by xi'an

From the current Riddler, a problem that only requires a few lines of code and a few seconds of reasoning. Or not.

N households each stole the earnings from one of the (N-1) other households, one at a time. What is the probability that a given household is not burglarised? And what are the expected final earnings of each household in the list, assuming they all start with \$1?

The first question is close to Feller’s enveloppe problem in that

$\left(1-\frac{1}{N-1}\right)^{N-1}$

is close to exp(-1) for N large. The second question can easily be solved by an R code like

N=1e3;M=1e6
fina=rep(1,N)
for (v in 1:M){
ordre=sample(1:N)
vole=sample(1:N,N,rep=TRUE)
while (min(abs(vole-(1:N)))==0)
vole[abs(vole-(1:N))==0]=sample(1:N,
sum(vole-(1:N)==0))
cash=rep(1,N)
for (t in 1:N){
cash[ordre[t]]=cash[ordre[t]]+cash[vole[t]];cash[vole[t]]=0}
fina=fina+cash[ordre]}


which returns a pretty regular exponential-like curve, although I cannot figure the exact curve beyond the third burglary. The published solution gives the curve

${\frac{N-2}{N-1}}^{999}\times 2+{\frac{1}{N-1}}^{t-1}\times{\frac{N-1}{N}}^{N-t}\times\frac{N}{N-1}$

corresponding to the probability of never being robbed (and getting on average an extra unit from the robbery) and of being robbed only before robbing someone else (with average wealth N/(N-1)).

## Le Monde puzzle [#1015]

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

A combinatoric Le Monde mathematical puzzle:

A game with N questions and N players is such that each question is solved by all players but three, while any pair of players fails to jointly solve the N questions. What is the maximal number N of players?

N=6;play=1;besz=1
while ((besz>0)||(play<N)){
gamz=matrix(1,N,N)
for (qez in 1:N) gamz[qez,sample(1:N,3)]=0
besz=0;play=1
while ((play<N)&(besz==0)){
besz=max(apply(as.matrix(gamz[,play]+
gamz[,(play+1):N],ncol=N-play),2,prod))
play=play+1}
}


The output of that code is N=7 in that N=8 does not escape the while loop. (Since there are 3N zeros to distribute in the NxN matrix, either all columns contain 3 zeros or one contains only two, in which case it can only share zeros with 2×2=4 other columns, which makes it impossible. When a column holds three zeros, it can share zeros with 3×2=6 other columns, which brings us back to N=7 as the highest possible case.)

A second game with M>N players and questions sees each question solved by all but four players. There is at least a pair of players jointly solving all questions. What is the minimal number M of players?

Given the simple update to the above R code

for (qez in 1:N) gamz[qez,sample(1:N,5)]=0


running the R code leads to suggest N=11, as the first instance when the loop does not exit. (The above logical argument does not run so well since having four zeros per column should allow for at most 4×3=12 other columns sharing zeros with that column, which leads to 14 as an upper bound for the answer, not 11!) However, the published solution is 14, which shows the limitation of this R code…

## Le Monde puzzle [#1014]

Posted in Books, Kids with tags , , , on June 30, 2017 by xi'an

One of those Le Monde puzzles where a computational solution seems out of reach: given 2N batteries out of which N are charged, how many tests of pairs of batteries are necessary to find a working pair? I first tried a brute force recursive resolution checking all (2N-1)N orders of pairs over all (2N)! permutations of 0…01…1, but the magnitude of the exploration is such that the program failed to conclude for N=3! After realising that N+4 is an upper bound on the number of attempts (since testing all N disjoint pairs until success or exhaustion leaves 4 attempts to find the two operative batteries within a couple of tested pairs), I managed to cap the depth of the recursion into the following R code:

library(combinat)
N=4
permz=permn(rep(c(0,1),N))
seqz=matrix(0,factorial(2*N),2*N)
for (i in 1:factorial(2*N)) seqz[i,]=permz[[i]]

optest=function(earl,scorz,N){
#remaining pairs
best=N+4
if (length(earl)>N+3) return(best)
for (i in 1:(2*N-1))
for (j in ((i+1):(2*N))){
prez=i+2*N*j
if (!(prez%in%earl)){
scorzz=apply(cbind(scorz,seqz[,i]*seqz[,j]),1,max)
if (min(scorzz)==1){
best=length(earl)+1
break()}else{
best=min(best,optest(c(earl,prez),scorzz,N))}}
if (best==6) break()
}
return(best)}

optest(earl=c(1+2*N*2,1+2*N*3),
scorz=apply(matrix(as.vector(seqz[,c(1,1)])*
as.vector(seqz[,c(2,3)]),ncol=2),1,max),N=N)


which returned a solution for N=3 (6 tests are enough, testing all combinations of the first three batteries then all combinations of the last three) but is still running for N=4! (I wonder at a possible resolution by modelling this puzzle though a very special type of multi-armed bandit problem with (pairwise) aggregated outcomes, but I did not go any further in that direction.)

## Le Monde puzzle [#1013]

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

A purely arithmetic Le Monde mathematical puzzle:

An operation þ applies to all pairs of natural integers with the properties

0 þ (a+1) = (0 þ a)+1, (a+1) þ (b+1)=(a þ b)+1, 271 þ 287 = 77777, 2018 þ 39 = 2018×39

Find the smallest integer d>287 such that there exists c<d leading to c þ d = c x d, the smallest integer f>2017 such that 2017 þ f = 2017×40. Is there any know integer f such that f þ 2017 = 40×2017?

The major appeal in this puzzle (where no R programming seems to help!) is that the “data” does not completely defines the operation  þ ! Indeed, when a<b, it is straightforward to deduce that a þ b = (0 þ 0)+b, hence solving the first two questions by deriving (0 þ 0)=270×287 [with d=315 and f=2017×40-270×287], but the opposed quantity b þ a is not defined, apart from (2018-39) þ 0. This however brings a resolution since

(2018-39) þ 0 = 2017×39 and (2018-39+2017) þ 2017 = 2017×39+2017 = 2017×40

## non-Bayesian decision riddle

Posted in Statistics with tags , , , on June 22, 2017 by xi'an

As a continuation of the Bayesian resolution of last week riddle, I looked at a numeric resolution of the four secretaries problem, while in the train back from Rouen (and trying to block the chatter of my neighbours, a nuisance I find myself more and more sensitive to!). The target function is defined as

gainz=function(b,c,T=1e4,type="raw"){
x=matrix(runif(4*T),ncol=4)
maz=t(apply(x,1,cummax))
zam=t(apply(x[,4:1],1,cummax))
if (type=="raw"){return(mean(
((x[,2]>b*x[,1])*x[,2]+
(x[,2]<b*x[,1])*((x[,3]>c*maz[,2])*x[,3]+
(x[,3]<c*maz[,2])*x[,4]))/maz[,4]))}
if (type=="global"){return(mean(
((x[,2]>b*x[,1])*(x[,2]==maz[,4])+
(x[,2]<b*x[,1])*((x[,3]>c*maz[,2])*(x[,3]==maz[,4])+
(x[,3]<c*maz[,2])*(x[,4]==maz[,4])))))}
if (type=="remain"){return(mean(
((x[,2]>b*x[,1])*(x[,2]==zam[,3])+
(x[,2]<b*x[,1])*((x[,3]>c*maz[,2])*(x[,3]==zam[,2])+
(x[,3]<c*maz[,2])*(x[,4]==zam[,2])))))}}


where the data is generated from a U(0,1) distribution as the loss functions are made scaled free by deciding to always sacrifice the first draw, x¹. This function is to be optimised in (b,c) and hence I used a plain vanilla simulated annealing R code:

avemale=function(T=3e4,type){
b=c=.5
maxtar=targe=gainz(b,c,T=1e4,type)
temp=0.1
for (t in 1:T){
bp=b+runif(1,-temp,temp)
cp=c+runif(1,-temp,temp)
parge=(bp>0)*(cp>0)*gainz(bp,cp,T=1e4,type)
if (parge>maxtar){
b=bs=bp;c=cs=cp;maxtar=targe=parge}else{
if (runif(1)<exp((parge-targe)/temp)){
b=bp;c=cp;targe=parge}}
temp=.9999*temp}
return(list(bs=bs,cs=cs,max=maxtar))}


with outcomes

• b=1, c=.5, and optimum 0.8 for the raw type
• b=c=1 and optimum 0.45 for the global type
• b undefined, c=2/3 and optimum 0.75 for the remain type

## Le Monde puzzle [#1012]

Posted in Books, Kids with tags , , , , , on June 14, 2017 by xi'an

A basic geometric Le Monde mathematical puzzle:

Take a triangle ABC such that the side AB is c=42 long, each side has an integer length, and the area is 756. Given an inner point D, draw three lines parallel to the three sides of ABC through D in order to construct three triangles with common summit D and bases supported by these three sides.

1. How far is D from the base AB when all three triangles have perimeters equal to the sides that support their basis?
2. How far is D from the previous solution when the sum of the areas of the three triangles is minimal?

Since the puzzle is purely geometric, I was quite tempted to bypass it and to watch instead the British elections and the Comey audition! However, the sides a and b are easily found by an exhaustive search, a=39 and b=45 (or the reverse). From there, the problem resolution proceeds by a similar triangles argument, since all triangles constructed by the game rule have the same angles, hence proportional sides. For the first question, this leads to a straightforward determination of the basis of each triangle by the perimeter equation, meaning that D is then 12 units away from AB. The second question is not harder in that the surface of a triangle with basis a and opposite angles β and γ can be written as

a²sin(β)sin(γ)/2sin(β+γ)

meaning it suffices to minimise a²+a’²+a”² under the constraint that the sum of the three sides parallel to BC is the complete length of BC, a²+a’²+a”²=39. The solution is then that all triangles are identical, leading to a summit D’ at a distance 12 from AB, again!, but in the middle of the segment, hence distance to the earlier D equal to one.