## 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
 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.

## Le Monde puzzle [#1045]

Posted in Books, Kids with tags , , , , , , on May 13, 2018 by xi'an

Take a sequence of 16  integers with 4 digits each, separated by 2,  such that it contains a perfect square and its sum is a perfect cube. What are the possible squares and cubes?

The question is dead easy to code in R

```for (x in as.integer(1e3:(1e4-16))){
if (max(round(sqrt(x+2*(0:15)))^2==x+2*(0:15))==1) {
b=sqrt((x+2*(0:15))[round(sqrt(x+2*(0:15)))^2==x+2*(0:15)])
if ((round((2*x+30)^(1/3)))^3==(2*x+30))
print(c(x,b,(16*(x+15))^(1/3)))}}
```

and return the following solutions:

``` 1357   37   28
 5309   73   44
```

Nothing that exciting…!