Carmichael number, more or less

A quick-and-dirty R resolution of a riddle from The Riddler, namely to find a Carmichael number of the form abcabc:

library(numbers)
for(i in 1:9)
  for(j in 0:9)
    for(k in 0:9){
      x=i*100100+j*1010+k*101
      if(!isPrime(x)){
        p=primeFactors(x)
        if((prod(apply(outer(p,p,F="=="),1,sum)%%2))&
          (!max((x-1)%%(p-1))))break()}}

resulting into the number 101 101, since its prime factors are

> primeFactors(101101)
[1]   7  11  13 101

and 6, 10, 12, and 100 are divisors of 101100:

> primeFactors(101100) 
[1] 2 2 3 5 5 337

easy and uneasy riddles

On 15 January, The Riddler had both a straightforward and a challenging riddles. The first one was to optimise the choice of a real number d with the utility function U(d,θ)=d ℑ(θ>d), when θ is Uniform(0,100). Leading unsurprisingly to d=50…

The tough(er) one was to solve a form of sudoku where the 24 entries of a 8×3 table are integers in {1,…,9} and the information is provided by the row-wise and column-wise products of these integers. The vertical margins are

294, 216, 135, 98, 112, 84, 245, 40

and the horizontal margins are

8 890 560, 156 800, 55 566

After an unsuccessful brute-force (and pseudo-annealed) attempt achieving a minimum error of 127, although using the prime factor decompositions of these 11 margins, I realised that some entries were known: e.g., 7 at (1,2), 5 at (3,2), and 7 at (7,3), and (much later) that the (huge) product value for the first column implied that each term in that column had to be the maximal possible value for the corresponding rows, except for 5 on row 7. This leads to the starting grid

    [,1] [,2] [,3]
[1,]    7    7    6
[2,]    9    0    0
[3,]    9    5    3
[4,]    7    0    0
[5,]    8    0    0
[6,]    7    0    0
[7,]    5    7    7
[8,]    8    0    0

and an additional and obvious exclusion based on the absence of 3’s in the second column, of 5’s and 2’s in the third column shows there was a unique solution

    [,1] [,2] [,3]
[1,]    7    7    6
[2,]    9    8    3
[3,]    9    5    3
[4,]    7    2    7
[5,]    8    2    7
[6,]    7    4    3
[7,]    5    7    7
[8,]    8    5    1

as also demonstrated by a complete exploration with R:

Try it online!

Fermat’s Riddle

·A Fermat-like riddle from the Riddler (with enough room to code on the margin)

An  arbitrary positive integer N is to be written as a difference of two distinct positive integers. What are the impossible cases and else can you provide a list of all distinct representations?

Since the problem amounts to finding a>b>0 such that

both (a+b) and (a-b) are products of some of the prime factors in the decomposition of N and both terms must have the same parity for the average a to be an integer. This eliminates decompositions with a single prime factor 2 (and N=1). For other cases, the following R code (which I could not deposit on tio.run because of the packages R.utils!) returns a list

library(R.utils)
library(numbers)
bitz<-function(i,m) #int2bits
  c(rev(as.binary(i)),rep(0,m))[1:m]
ridl=function(n){
a=primeFactors(n)
if((n==1)|(sum(a==2)==1)){
  print("impossible")}else{
  m=length(a);g=NULL
  for(i in 1:2^m){
    b=bitz(i,m)
    if(((d<-prod(a[!!b]))%%2==(e<-prod(a[!b]))%%2)&(d<e))
      g=rbind(g,c(k<-(e+d)/2,l<-(e-d)/2))}
  return(g[!duplicated(g[,1]-g[,2]),])}}

For instance,

> ridl(1456)
     [,1] [,2]
[1,]  365  363
[2,]  184  180
[3,]   95   87
[4,]   59   45
[5,]   40   12
[6,]   41   15

Checking for the most prolific N, up to 10⁶, I found that N=6720=2⁶·3·5·7 produces 20 different decompositions. And that N=887,040=2⁸·3²·5·7·11 leads to 84 distinct differences of squares.

riddle of the seats

An arithmetic quick riddle from The Riddler:

If an integer n is a multiple of every integer between 1 and 200, except for two consecutive ones, find those consecutive integers.

Since the highest power of 2 less than 200 is 2⁷=128 and since 127 is a prime number, the number

should work in that it contains all odd integers but 127, and all even numbers, but 128. Of course a smaller number that avoids duplicates by only considering the 44 primes other than 127 and 2 to a power that keep them less than 200 is also valid. Which gives a number of the order of 1.037443 10⁸⁵.

missing digit in a 114 digit number [a Riddler’s riddle]

A puzzling riddle from The Riddler (as Le Monde had a painful geometry riddle this week): this number with 114 digits

530,131,801,762,787,739,802,889,792,754,109,70?,139,358,547,710,066,257,652,050,346,294,484,433,323,974,747,960,297,803,292,989,236,183,040,000,000,000

is missing one digit and is a product of some of the integers between 2 and 99. By comparison, 76! and 77! have 112 and 114 digits, respectively. While 99! has 156 digits. Using WolframAlpha on-line prime factor decomposition code, I found that only 6 is a possible solution, as any other integer between 0 and 9 included a large prime number in its prime decomposition:

However, I thought anew about it when swimming the next early morning [my current substitute to morning runs] and reasoned that it was not necessary to call a formal calculator as it is reasonably easy to check that this humongous number has to be divisible by 9=3×3 (for else there are not enough terms left to reach 114 digits, checked by lfactorial()… More precisely, 3³³x33! has 53 digits and 99!/3³³x33! 104 digits, less than 114), which means the sum of all digits is divisible by 9, which leads to 6 as the unique solution.