## Le Monde puzzle [#990]

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , on January 12, 2017 by xi'an To celebrate the new year (assuming it is worth celebrating!), Le Monde mathematical puzzle came up with the following:

Two sequences (x¹,x²,…) and (y¹,y²,…) are defined as follows: the current value of x is either the previous value or twice the previous value, while the current value of y is the sum of the values of x up to now. What is the minimum number of steps to reach 2016 or 2017?

By considering that all consecutive powers of 2 must appear at least one, the puzzles boils down to finding the minimal number of replications in the remainder of the year minus the sum of all powers of 2. Which itself boils down to deriving the binary decomposition of that remainder. Hence the basic R code (using intToBits):

```deco=function(k=2016){
m=trunc(log2(k))
while (sum(2^(0:m))>k) m=m-1
if (sum(2^(0:m))==k){ return(rep(1,m+1))
}else{
res=k-sum(2^(0:m))
return(rep(1,m+1)+as.integer(intToBits(res))[1:(m+1)])
```

which produces

```> sum(deco(2016))
 16
> sum(deco(2017))
 16
> sum(deco(1789))
 18
```

## Le Monde sans puzzle [& sans penguins]

Posted in Books, Kids, R, University life with tags , , , , , on April 12, 2014 by xi'an As the Le Monde mathematical puzzle of this week was a geometric one (the quadrangle ABCD is divided into two parts with the same area, &tc…) , with no clear R resolution, I chose to bypass it. In this April 3 issue, several items of interest: first, a report by Etienne Ghys on Yakov Sinaï’s Abel Prize for his work “between determinism and randomness”, centred on ergodic theory for dynamic systems, which sounded like the ultimate paradox the first time I heard my former colleague Denis Bosq give a talk about it in Paris 6. Then a frightening fact: the summer conditions have been so unusually harsh in Antarctica (or at least near the Dumont d’Urville French austral station) that none of the 15,000 Adélie penguin couples studied there managed to keep their chick alive. This was due to an ice shelf that did not melt at all over the summer, forcing the penguins to walk an extra 40k to reach the sea… Another entry on the legal obligation for all French universities to offer a second chance exam, no matter how students are evaluated in the first round. (Too bad, I always find writing a second round exam a nuisance.)

## Le Monde puzzle [#843]

Posted in Books, Kids, R with tags , , , , , on December 7, 2013 by xi'an A Le Monde mathematical puzzle of moderate difficulty:

How many binary quintuplets (a,b,c,d,e) can be found such that any pair of quintuplets differs by at least two digits?

I solved it by the following R code that iteratively eliminates quintuplets that are not different enough from the first ones, for a random order of the 2⁵ quintuplets because the order matters in the resulting number (the intToBits trick was provided by an answer on StackExchange/stackoverflow):

```sol=0
for (t in 1:10^5){ #random permutations
as.integer(intToBits(x))})[1:5,sample(1:32)]
V=32;inin=rep(TRUE,V);J=1
while (J<V){
for (i in (J+1):V)
inin[i]=FALSE
J=J+1}
if (sol<V){
}
```

which returns solutions like

```> sol
 16
> levote
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
[1,]  0    0    0    0    1    1    1    1    0     1     0
[2,]  0    1    0    1    0    1    0    1    0     1     1
[3,]  0    1    1    0    1    0    1    1    1     0     0
[4,]  0    1    1    1    0    0    0    0    0     1     0
[5,]  0    0    0    0    0    0    1    0    0     0     0
[,12] [,13] [,14] [,15] [,16]
[1,]    0    1     1     0     1
[2,]    0    1     1     0     1
[3,]    1    0     0     1     1
[4,]    0    0     1     1     0
[5,]    1    0     1     0     1
```