Archive for mathematical puzzle

Le Monde puzzle [#1104]

Posted in Kids, R with tags , , , , on June 18, 2019 by xi'an

A palindromic Le Monde mathematical puzzle:

In a monetary system where all palindromic amounts between 1 and 10⁸ have a coin, find the numbers less than 10³ that cannot be paid with less than three coins. Find if 20,191,104 can be paid with two coins. Similarly, find if 11,042,019 can be paid with two or three coins.

Which can be solved in a few lines of R code:

coin=sort(c(1:9,(1:9)*11,outer(1:9*101,(0:9)*10,"+")))
amounz=sort(unique(c(coin,as.vector(outer(coin,coin,"+")))))
amounz=amounz[amounz<1e3]

and produces 9 amounts that cannot be paid with one or two coins.

21 32 43 54 65 76 87 98 201

It is also easy to check that three coins are enough to cover all amounts below 10³. For the second question, starting with n¹=20,188,102,  a simple downward search of palindromic pairs (n¹,n²) such that n¹+n²=20,188,102 led to n¹=16,755,761 and n²=3,435,343. And starting with 11,033,011, the same search does not produce any solution, while there are three coins such that n¹+n²+n³=11,042,019, for instance n¹=11,022,011, n²=20,002, and n³=6.

easy Riddler

Posted in Kids, R with tags , , , on May 10, 2019 by xi'an

The riddle of the week is rather standard probability calculus

If N points are generated at random places on the perimeter of a circle, what is the probability that you can pick a diameter such that all of those points are on only one side of the newly halved circle?

Since it is equivalent to finding the range of N Uniform variates less than ½. And since the range of N Uniform variates is distributed as a Be(N-1,2) random variate. The resulting probability, which happens to be exactly N/2^{N-1}, is decreasing exponentially, as shown below…

Le Monde puzzle [#1099]

Posted in Books, Kids, R with tags , , , , , on April 28, 2019 by xi'an

A simple 2×2 Le Monde mathematical puzzle:

Arielle and Brandwein play a game out of two distinct even integers between 1500 and 2500,  and y. Providing one another with either the pair (x/2,y+x/2) or the pair (x+y/2,y/2) until they run out of even possibilities or exceed 6 rounds. When x=2304, what is the value of y that makes Brandwein win?

Which I solved by a recursive function (under the constraint of a maximum of 11 levels of recursion):

nezt=function(x,y,i=1){
  if ((i>11)||((is.odd(x)&is.odd(y)))){ return(-1)
  }else{
    z=-1
    if (is.even(x)) z=-nezt(x/2,y+x/2,i+1)
    if (is.even(y)) z=max(z,-nezt(y/2,x+y/2,i+1))
    return(z)}}

and checking all values of y between 1500 and 2500 when x=2304, which produces y=1792 as the only value when Arielle loses. The reason behind (?) is that both 2304 and 1792 are divisible by 2⁸, which means no strategy avoids reaching stalemate after 8 steps, when it is Arielle’s turn to play.

Le Monde puzzle [#1094]

Posted in Books, Kids, R with tags , , , , , , on April 23, 2019 by xi'an

A rather blah number Le Monde mathematical puzzle:

Find all integer multiples of 11111 with exactly one occurrence of each decimal digit..

Which I solved by brute force, by looking at the possible range of multiples (and  borrowing stringr:str_count from Robin!)

> combien=0
> for (i in 90001:900008){
  j=i*11111
  combien=combien+(min(stringr::str_count(j,paste(0:9)))==1)}
> combien
[1] 3456

And a bonus one:

Find all integers y that can write both as x³ and (10z)³+a with 1≤a≤999.

which does not offer much in terms of solutions since x³-v³=(x-v)(x²+xv+v²)=a shows that x² is less than 2a/3, meaning x is at most 25. Among such numbers only x=11,12 lead to a solution as x³=1331,1728.

survivalists [a Riddler’s riddle]

Posted in Books, Kids, R, Statistics with tags , , , , , , on April 22, 2019 by xi'an

A neat question from The Riddler on a multi-probability survival rate:

Nine processes are running in a loop with fixed survivals rates .99,….,.91. What is the probability that the first process is the last one to die? Same question with probabilities .91,…,.99 and the probability that the last process is the last one to die.

The first question means that the realisation of a Geometric G(.99) has to be strictly larger than the largest of eight Geometric G(.98),…,G(.91). Given that the cdf of a Geometric G(a) is [when counting the number of attempts till failure, included, i.e. the Geometric with support the positive integers]

F(x)=\Bbb P(X\le x)=1-a^{x}

the probability that this happens has the nice (?!) representation

\sum_{x=2}^\infty a_1^{x-1}(1-a_1)\prod_{j\ge 2}(1-a_j^{x-1})=(1-a_1)G(a_1,\ldots,a_9)

which leads to an easy resolution by recursion since

G(a_1,\ldots,a_9)=G(a_1,\ldots,a_8)-G(a_1a_9,\ldots,a_8)

and G(a)=a/(1-a)

and a value of 0.5207 returned by R (Monte Carlo evaluation of 0.5207 based on 10⁷ replications). The second question is quite similar, with solution

\sum_{x=2}^\infty a_1^{x-1}(1-a_1)\prod_{j\ge 1}(1-a_j^{x})=a^{-1}(1-a_1)G(a_1,\ldots,a_9)

and value 0.52596 (Monte Carlo evaluation of 0.52581 based on 10⁷ replications).

Le Monde puzzle [#1092]

Posted in Statistics with tags , , , , , , , on April 18, 2019 by xi'an

A Latin square Le Monde mathematical puzzle that I found rather dreary:

A hidden 3×3 board contains all numbers from 1 to 9. Anselm wants to guess the board and makes two proposals. Berenicke tells him how many entries are in the right rows and colums for each proposal, along with the information that no entry is at the right location. Anselm deduces the right board.

Which I solved by brute force and not even simulated annealing, first defining a target

ordoku1=ordoku2=matrix(1,9,2)
ordoku1[,1]=c(1,1,1,2,2,2,3,3,3)
ordoku1[,2]=rep(1:3,3)
ordoku2[,1]=c(3,2,3,1,2,3,2,1,1)
ordoku2[,2]=c(2,2,3,2,3,1,1,3,1)
fitz=function(ordo){
  (sum(ordo[c(1,4,7),2]==1)==1)+(sum(ordo[c(2,5,8),2]==2)==1)+
  (sum(ordo[c(3,6,9),2]==3)==0)+(sum(ordo[c(1,2,3),1]==1)==1)+
  (sum(ordo[c(4,5,6),1]==2)==1)+(sum(ordo[c(7,8,9),1]==3)==2)+
  (sum(ordo[c(6,7,9),2]==1)==2)+(sum(ordo[c(1,2,4),2]==2)==1)+
  (sum(ordo[c(3,5,8),2]==3)==2)+(sum(ordo[c(4,8,9),1]==1)==1)+
  (sum(ordo[c(7,2,5),1]==2)==1)+(sum(ordo[c(1,3,6),1]==3)==0)+
  (!(0%in%apply((ordo-ordoku1)^2,1,sum)))+(!(0%in%apply((ordo-ordoku2)^2,1,sum)))
}

on a 9×9 board entry reproducing all items of information given by Berenicke. If all constraints are met, the function returns 14. And then searched for a solution at random:

temp=1
randw=function(ordo){
  for (t in 1:1e6){
    chlg=sample(1:9,2)
    temp=ordo[chlg[1],]
    ordo[chlg[1],]=ordo[chlg[2],]
    ordo[chlg[2],]=temp
    if (fitz(ordo)==14){
      print(ordo);break()}}}

which produces the correct board

4 3 5
6 7 1
9 2 8

no country for old liars

Posted in Kids, R with tags , , , , , on March 30, 2019 by xi'an

A puzzle from the Riddler about a group of five persons, A,..,E, where all and only people strictly older than L are liars, all making statements about others’ ages:

  1. A: B>20 and D>16
  2. B: C>18 and E<20
  3. C: D<22 and A=19
  4. D: E≠20 and B=20
  5. E: A>21 and C<18

The Riddler is asking for the (integer value of L and the ranges or values of A,…,E. After thinking about this puzzle over a swimming session, I coded the (honest) constraints and their (liar) complements as many binary matrices, limiting the number of values of L to 8 from 0 (15) to 7 (22) and A,…,E to 7 from 1 (16) to 7 (22):

CA=CB=CC=CD=CE=A=B=C=D=E=matrix(1,5,7)
#constraints
A[2,1:(20-15)]=A[4,1]=0 #A honest
CA[2,(21-15):7]=CA[4,2:7]=0 #A lying
B[3,1:(18-15)]=B[5,(20-15):7]=0
CB[3,(19-15):7]=CB[5,1:(19-15)]=0
C[1,-(19-15)]=C[4,7]=0 #C honest
CC[1,(19-15)]=CC[4,-7]=0 #C lying
D[5,(17-15)]=D[2,-(20-15)]=0
CD[5,-(17-15)]=CD[2,(20-15)]=0
E[1,1:(21-15)]=E[3,(18-15):7]=0
CE[1,7]=CE[3,1:(17-15)]=0

since the term-wise product of these five matrices expresses all the constraints on the years, as e.g.

ABCDE=A*CB*CC*D*CE

if A,D≤L and B,C,E>L, and I then looked by uniform draws [with a slight Gibbs flavour] for values of the integers that suited the constraints or their complement, the stopping rule being that the collection of A,…,E,L is producing an ABCDE binary matrix that agrees with all statements modulo the lying statuum of their authors:

yar=1:5
for (i in 1:5) yar[i]=sample(1:7,1)
L=sample(0:7,1)
ABCDE=((yar[1]>L)*CA+(yar[1]<=L)*A)* 
   ((yar[2]>L)*CB+(yar[2]<=L)*B)* 
   ((yar[3]>L)*CC+(yar[3]<=L)*C)* 
   ((yar[4]>L)*CD+(yar[4]<=L)*D)* 
   ((yar[5]>L)*CE+(yar[5]<=L)*E) 
while (min(diag(ABCDE[,yar]))==0){ 
   L=sample(0:7,1);idx=sample(1:5,1) 
   if (max(ABCDE[idx,])==1) yar[idx]=sample(which(ABCDE[idx,]>0),1)
   ABCDE=((yar[1]>L)*CA+(yar[1]<=L)*A)* 
   ((yar[2]>L)*CB+(yar[2]<=L)*B)* 
   ((yar[3]>L)*CC+(yar[3]<=L)*C)*
   ((yar[4]>L)*CD+(yar[4]<=L)*D)* 
   ((yar[5]>L)*CE+(yar[5]<=L)*E) 
    }

which always produces L=18,A=19,B=20,C=18,D=16 and E>19 as the unique solution (also reported by The Riddler).

> ABCDE
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,]    0    0    0    1    0    0    0
[2,]    0    0    0    0    1    0    0
[3,]    0    0    1    0    0    0    0
[4,]    1    0    0    0    0    0    0
[5,]    0    0    0    0    1    1    1