Archive for arithmetics

Le Monde puzzle [#1083]

Posted in Books, Kids, R, Travel with tags , , , , , , on February 7, 2019 by xi'an

A Le Monde mathematical puzzle that seems hard to solve without the backup of a computer (and just simple enough to code on a flight to Montpellier):

Given the number N=2,019, find a decomposition of N as a sum of non-trivial powers of integers such that (a) the number of integers in the sum is maximal or (b) all powers are equal to 4.  Is it possible to write N as a sum of two powers?

It is straightforward to identify all possible terms in these sums by listing all powers of integers less than N

pool=(1:trunc(sqrt(2019)))^2
for (pow in 3:11)
  pool=unique(c(pool,(2:trunc(2019^(1/pow)))^pow))

which leads to 57 distinct powers. Sampling at random from this collection at random produces a sum of 21 perfect powers:

 1+4+8+9+16+25+27+32+36+49+64+81+100+121+125+128+144+169+196+243+441

But looking at the 22 smallest numbers in the pool of powers leads to 2019, which is a sure answer. Restricting the terms to powers of 4 leads to the sequence

1⁴+2⁴+3⁴+5⁴+6⁴ = 2019

And starting from the pools of all possible powers in a decomposition of 2019 as the sum of two powers shows this is impossible.

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 [#1063]

Posted in Books, Kids, R with tags , , , , , , on August 9, 2018 by xi'an

lemondapariA simple (summertime?!) arithmetic Le Monde mathematical puzzle

  1. A “powerful integer” is such that all its prime divisors are at least with multiplicity 2. Are there two powerful integers in a row, i.e. such that both n and n+1 are powerful?
  2.  Are there odd integers n such that n² – 1 is a powerful integer ?

The first question can be solved by brute force.  Here is a R code that leads to the solution:

isperfz <- function(n){ 
  divz=primeFactors(n) 
  facz=unique(divz) 
  ordz=rep(0,length(facz)) 
  for (i in 1:length(facz)) 
    ordz[i]=sum(divz==facz[i]) 
  return(min(ordz)>1)}

lesperf=NULL
for (t in 4:1e5)
if (isperfz(t)) lesperf=c(lesperf,t)
twinz=lesperf[diff(lesperf)==1]

with solutions 8, 288, 675, 9800, 12167.

The second puzzle means rerunning the code only on integers n²-1…

[1] 8
[1] 288
[1] 675
[1] 9800
[1] 235224
[1] 332928
[1] 1825200
[1] 11309768

except that I cannot exceed n²=10⁸. (The Le Monde puzzles will now stop for a month, just like about everything in France!, and then a new challenge will take place. Stay tuned.)

seven digit addition

Posted in Kids, R with tags , , , on July 6, 2018 by xi'an

Another quick riddle from the riddler: solve the equation

EXMREEK + EHKREKK = ?K?H?X?E

which involves every digit between 0 and 9. While the puzzle can be unravelled by considering first E and K, which must be equal to 6 and 3, a simple R code also leads to the conclusion

isok <- function(a,b){
 s=as.numeric(unlist(strsplit(as.character(sum(10^(6:0)*a)+
   sum(10^(6:0)*b)),"")))
 if (length(s)==7){ goal=FALSE}else{
   goal=(length(unique(c(a,b,s)))==10)&(a[2]==s[6])&
         (s[8]==a[6])&(s[2]==a[7])&(b[2]==s[4])}
 return(goal)}

pasok <- function(T=1e3){ 
for (t in 1:T){ 
  a[1]=a[5]=a[6]=6;a[7]=3 
  digs=sample(c(0:2,4,5,7:9),4) 
  a[2:4]=digs[1:3] b[1]=a[1];b[2]=digs[4];
  b[3]=a[7];b[4]=a[4];b[5]=a[1];b[6:7]=a[7] 
  if (isok(a=a,b=b)) 
     print(rbind(a,b))}} 

> pasok()
  [,1] [,2] [,3] [,4] [,5] [,6] [,7]
a    6    2    4    7    6    6    3
b    6    8    3    7    6    3    3

which sum is 13085296.

Le Monde puzzle [#1018]

Posted in Books, Kids, R with tags , , , , , on August 29, 2017 by xi'an

An arithmetic Le Monde mathematical puzzle (that first did not seem to involve R programming because of the large number of digits in the quantity involved):

An integer x with less than 100 digits is such that adding the digit 1 on both sides of x produces the integer 99x.  What are the last nine digits of x? And what are the possible numbers of digits of x?

The integer x satisfies the identity

10^{\omega+2}+10x+1=99x

where ω is the number of digits of x. This amounts to

10….01 = 89 x,

where there are ω zeros. Working with long integers in R could bring an immediate solution, but I went for a pedestrian version, handling each digit at a time and starting from the final one which is necessarily 9:

#multiply by 9
rap=0;row=NULL
for (i in length(x):1){
prud=rap+x[i]*9
row=c(prud%%10,row)
rap=prud%/%10}
row=c(rap,row)
#multiply by 80
rep=raw=0
for (i in length(x):1){
prud=rep+x[i]*8
raw=c(prud%%10,raw)
rep=prud%/%10}
#find next digit
y=(row[1]+raw[1]+(length(x)>1))%%10

returning

7 9 7 7 5 2 8 0 9

as the (only) last digits of x. The same code can be exploited to check that the complete multiplication produces a number of the form 10….01, hence to deduce that the length of x is either 21 or 65, with solutions

[1] 1 1 2 3 5 9 5 5 0 5 6 1 7 9 7 7 5 2 8 0 9
[1] 1 1 2 3 5 9 5 5 0 5 6 1 7 9 7 7 5 2 8 0 8 9 8 8 7 6 4 0 4 4 9 4 3 8 2 0 2 2
[39] 4 7 1 9 1 0 1 1 2 3 5 9 5 5 0 5 6 1 7 9 7 7 5 2 8 0 9

The maths question behind is to figure out the powers k of 10 such that

10^k\equiv -1 \text{ mod } (89)

For instance, 10²≡11 mod (89) and 11¹¹≡88 mod (89) leads to the first solution ω=21. And then, since 10⁴⁴≡1 mod (89), ω=21+44=65 is another solution…

continental divide

Posted in Books, Kids, pictures, R with tags , , , , on May 19, 2017 by xi'an

While the Riddler puzzle this week was anticlimactic,  as it meant filling all digits in the above division towards a null remainder, it came as an interesting illustration of how different division is taught in the US versus France: when I saw the picture above, I had to go and check an American primary school on-line introduction to division, since the way I was taught in France is something like that

with the solution being that 12128316 = 124 x 97809… Solved by a dumb R exploration of all constraints:

for (y in 111:143)
for (z4 in 8:9)
for (oz in 0:999){
  z=oz+7e3+z4*1e4
  x=y*z
  digx=digits(x)
  digz=digits(z)
  if ((digz[2]==0)&(x>=1e7)&(x<1e8)){ 
   r1=trunc(x/1e4)-digz[5]*y 
   if ((digz[5]*y>=1e3)&(digz[4]*y<1e4) &(r1>9)&(r1<100)){ 
    r2=10*r1+digx[4]-7*y 
    if ((7*y>=1e2)&(7*y<1e3)&(r2>=1e2)&(r2<1e3)){     
     r3=10*r2+digx[3]-digz[3]*y 
     if ((digz[3]*y>=1e2)&(digz[3]*y<1e3)&(r3>9)&(r3<1e2)){
       r4=10*r3+digx[2]
       if (r4<y) solz=rbind(solz,c(y,z,x))
  }}}}

Looking for a computer-free resolution, the constraints on z exhibited by the picture are that (a) the second digit is 0 and the fourth digit is 7.  Moreover, the first and fifth digits are larger than 7 since y times these digits is a four-digit number. Better, since the second subtraction from a three-digit number by 7y returns a three-digit number and the third subtraction from a four-digit number by ny returns a two-digit number, n is larger than 7 but less than the first and fifth digits. Ergo, z is necessarily 97809! Furthermore, 8y<10³ and 9y≥10³, which means 111<y<125. Plus the constraint that 1000-8y≤99 implies y≥112. Nothing gained there! This leaves 12 values of y to study, unless there is another restriction I missed…

Le Monde puzzle [#1000…1025]

Posted in Kids, R with tags , , , , , , on March 28, 2017 by xi'an

Le Monde mathematical puzzle launched a competition to celebrate its 1000th puzzle! A fairly long-term competition as it runs over the 25 coming puzzles (and hence weeks). Starting with puzzle #1001. Here is the 1000th puzzle, not part of the competition:

Alice & Bob spend five (identical) vouchers in five different shops, each time buying the maximum number of items to get close to the voucher value. In these five shops, they buy sofas at 421 euros each, beds at 347 euros each, kitchen appliances at 289 euros each, tables at 251 euros each and bikes at 211 euros each, respectively. Once the buying frenzy is over, they realise that within a single shop, they would have spent exactly four vouchers for the same products. What is the value of a voucher?