## Le Monde puzzle [#1076]

Posted in Books, Kids, R, Travel with tags , , , , , , , , , on December 27, 2018 by xi'an

A cheezy Le Monde mathematical puzzle : (which took me much longer to find [in the sense of locating] than to solve, as Warwick U does not get a daily delivery of the newspaper [and this is pre-Brexit!]):

Take a round pizza (or a wheel of Gruyère) cut into seven identical slices and turn one slice upside down. If the only possibly moves are to turn three connected slices to their reverse side, how many moves at least are needed to recover the original configuration? What is the starting configuration that requires the largest number of moves?

Since there are ony N=2⁷ possible configurations, a brute force exploration is achievable, starting from the perfect configuration requiring zero move and adding all configurations found by one additional move at a time… Until all configurations have been visited and all associated numbers of steps are stable. Here is my R implementation

nztr=lengz=rep(-1,N) #length & ancestor
nztr[0+1]=lengz[0+1]=0
fundz=matrix(0,Z,Z) #Z=7
for (i in 1:Z){ #only possible moves
fundz[i,c(i,(i+1)%%Z+Z*(i==(Z-1)),(i+2)%%Z+Z*(i==(Z-2)))]=1
lengz[bit2int(fundz[i,])+1]=1
nztr[bit2int(fundz[i,])+1]=0}
while (min(lengz)==-1){ #second loop omitted
for (j in (1:N)[lengz>-1])
for (k in 1:Z){
m=bit2int((int2bit(j-1)+fundz[k,])%%2)+1
if ((lengz[m]==-1)|(lengz[m]>lengz[j]+1)){
lengz[m]=lengz[j]+1;nztr[m]=j}
}}


Which produces a path of length five returning (1,0,0,0,0,0,0) to the original state:

> nztry(2)
[1] 1 0 0 0 0 0 0
[1] 0 1 1 0 0 0 0
[1] 0 1 0 1 1 0 0
[1] 0 1 0 0 0 1 0
[1] 1 1 0 0 0 0 1
[1] 0 0 0 0 0 0 0


and a path of length seven in the worst case:

> nztry(2^7)
[1] 1 1 1 1 1 1 1
[1] 1 1 1 1 0 0 0
[1] 1 0 0 0 0 0 0
[1] 0 1 1 0 0 0 0
[1] 0 1 0 1 1 0 0
[1] 0 1 0 0 0 1 0
[1] 1 1 0 0 0 0 1
[1] 0 0 0 0 0 0 0


Since the R code was written for an arbitrary number Z of slices, I checked that there is no solution for Z being a multiple of 3.

## running plot [and simulated annealing]

Posted in Kids, pictures, R with tags , , on December 14, 2018 by xi'an

Last weekend, I found out a way to run updated plots within a loop in R, when calling plot() within the loop was never updated in real time. The above suggestion of including a Sys.sleep(0.25) worked perfectly on a simulated annealing example for determining the most dispersed points in a unit disc.

## Le Monde puzzle [#1075]

Posted in Books, Kids, R with tags , , , , on December 12, 2018 by xi'an

A new Le Monde mathematical puzzle in the digit category:

Find the largest number such that each of its internal digits is strictly less than the average of its two neighbours. Same question when all digits differ.

For instance, n=96433469 is such a number. When trying pure brute force (with the usual integer2digits function!)

le=solz=3
while (length(solz)>0){
solz=NULL
for (i in (10^(le+1)-1):(9*10^le+9)){
x=as.numeric(strsplit(as.character(i), "")[[1]])
if (min(x[-c(1,le+1)]<(x[-c(1,2)]+x[-c(le,le+1)])/2)==1){ print(i);solz=c(solz,i); break()}}
le=le+1}


this is actually the largest number returned by the R code. There is no solution with 9 digits. Adding an extra condition

le=solz=3
while (length(solz)>0){
solz=NULL
for (i in (10^(le+1)-1):(9*10^le+9)){
x=as.numeric(strsplit(as.character(i), "")[[1]])
if ((min(x[-c(1,le+1)]<(x[-c(1,2)]+x[-c(le,le+1)])/2)==1)&
(length(unique(x))==le+1)){ print(i);solz=c(solz,i); break()}}
le=le+1}


produces n=9520148 (seven digits) as the largest possible integer.

## Le Monde puzzle [#1075]

Posted in Books, Kids, R with tags , , , , , , , on November 22, 2018 by xi'an

A Le Monde mathematical puzzle from after the competition:

A sequence of five integers can only be modified by subtracting an integer N from two neighbours of an entry and adding 2N to the entry.  Given the configuration below, what is the minimal number of steps to reach non-negative entries everywhere? Is this feasible for any configuration?

As I quickly found a solution by hand in four steps, but missed the mathematical principle behind!, I was not very enthusiastic in trying a simulated annealing version by selecting the place to change inversely proportional to its value, but I eventually tried and also obtained the same solution:

      [,1] [,2] [,3] [,4] [,5]
-3    1    1    1    1
1   -1    1    1   -1
0    1    0    1   -1
-1    1    0    0    1
1    0    0    0    0


But (update!) Jean-Louis Fouley came up with one step less!

      [,1] [,2] [,3] [,4] [,5]
-3    1    1    1    1
3   -2    1    1   -2
2    0    0    1   -2
1    0    0    0    0


The second part of the question is more interesting, but again without a clear mathematical lead, I could only attempt a large number of configurations and check whether all admitted “solutions”. So far none failed.

## out of desperation

Posted in Books, Kids, Statistics, University life with tags , , , on November 9, 2018 by xi'an

Someone desperately seeking solutions to the even numbered questions of Introducing Monte Carlo Methods with R…. How odd!

## Le Monde puzzle [#1073]

Posted in Books, Kids, R with tags , , , , , , on November 3, 2018 by xi'an

And here is Le Monde mathematical puzzle  last competition problem

Find the number of integers such that their 15 digits are all between 1,2,3,4, and the absolute difference between two consecutive digits is 1. Among these numbers how many have 1 as their three-before-last digit and how many have 2?

Combinatorics!!! While it seems like a Gordian knot because the number of choices depends on whether or not a digit is an endpoint (1 or 4), there is a nice recurrence relation between the numbers of such integers with n digits and leftmost digit i, namely that

n⁴=(n-1)³, n³=(n-1)²+(n-1)⁴, n²=(n-1)²+(n-1)¹, n¹=(n-1)²

with starting values 1¹=1²=1³=1⁴=1 (and hopefully obvious notations). Hence it is sufficient to iterate the corresponding transition matrix

$M= \left(\begin{matrix}0 &1 &0 &0\\1 &0 &1 &0\\0 &1 &0 &1\\0 &0 &1 &0\\\end{matrix} \right)$

on this starting vector 14 times (with R, which does not enjoy a built-in matrix power) to end up with

15¹=610, 15²= 987, 15³= 987, 15⁴= 610

which leads to 3194 different numbers as the solution to the first question. As pointed out later by Amic and Robin in Dauphine, this happens to be twice Fibonacci’s sequence.

For the second question, the same matrix applies, with a different initial vector. Out of the 3+5+5+3=16 different integers with 4 digits, 3 start with 1 and 5 with 2. Then multiplying (3,0,0,0) by M¹⁰ leads to 267+165=432 different values for the 15 digit integers and multiplying (0,5,0,0) by M¹⁰ to. 445+720=1165 integers. (With the reassuring property that 432+1165 is half of 3194!) This is yet another puzzle in the competition that is of no special difficulty and with no further challenge going from the first to the second question…

## Le Monde puzzle [#1072]

Posted in Books, Kids, R with tags , , , , , , , , , , , on October 31, 2018 by xi'an

The penultimate Le Monde mathematical puzzle  competition problem is once again anti-climactic and not worth its points:

For the figure below [not the original one!], describing two (blue) half-circles intersecting with a square of side 20cm, and a (orange) quarter of a circle with radius 20cm, find the radii of both gold circles, respectively tangent to both half-circles and to the square and going through the three intersections.

Although the problem was easily solvable by some basic geometry arguments, I decided to use them a minima and resort to a mostly brute-force exploration based on a discretisation of the 20×20 square, from which I first deduced the radius of the tangent circle by imposing (a) a centre O on the diagonal (b) a distance from O to one (half-)circle equal to the distance to the upper side of the square, leading to a radius of 5.36 (and a centre O at a distance 20.70 from the bottom left corner):

diaz=sqrt(2)*20
for (diz in seq(5/8,7/8,le=1e4)*diaz){#position of O

for (diz in seq(20*sqrt(2)-20,10*sqrt(2),le=1e4)){