Problems & Puzzles: Puzzles

Puzzle 1027. Integers as sum of distinct repdigits Rodolfo Kurchan sent the following nice puzzle We have up to nine repdigits numbers, from 1 to 9. Each number can have 0<=N<10 equal digits. It is invalid to add two integers with the same digit even if these two have distinct quantity of digits. Example 8888 y 888 can not be used in the same expression.   Example with a solution: 98765 = 88888 + 7777 + 1111 + 555 + 333 + 99 + 2 On the other hand, I think that 987654 o 987650 are impossible to be expressed as said above. Q1. What is the minimal integer impossible to be expresses as said. Q2. Redo Q1 for prime numbers. Q3. Redo Q1 for zero-free pandigitals In any case show your solutions for the three numbers of each type previous to the one without solution

---

From Jan 9-15, 2021, contributions came from Emmanuel Vantieghem, Paul Cleary, Oscar Volpatti

***

Emmanuel wrote:

Q1.
 

The smallest number not representable as a sum of repdigits is  25427.
Solutions for the three previous numbers are :
   25424 = 11111 + 222 + 3333 + 5 + 666 + 88 + 9999
   25425 = 11111 + 2222 + 3333 + 444 + 555 + 6666 + 7 + 88 + 999
   25426 = 11111 + 2222 + 333 + 4444 + 555 + 6666 + 7 + 88

Q2.
The smallest prime not representable as a sum of repdigits is  32027.
Solutions for the three previous prime are :
   31991 = 11111 + 2222 + 3333 + 5555 + 6 + 777 + 8888 + 99
   32003 = 11111 + 2222 + 3333 + 4444 + 6 + 888 + 9999
   32009 = 11111 + 2222 + 3333 + 4444 + 5 + 7 + 888 + 9999

 

Q3.
The smallest zero-free pandigital is  123456789  and it has no representation as a sum of repdigits.

 

(Not asked : the smallest representable zero-free pandigital is :
   123457896 = 11111 + 2222 + 33 + 444444 + 55555555 + 66666666 + 777777 + 88
 and the biggest :  
     984673251 = 11111111 + 222222 + 33 + 5555 + 6666666 + 77777777 + 888888888 + 999)

***

Paul wrote:

Q1.

 The minimum number is 25427.

25424 = 2 + 33 + 4 + 55 + 6666 + 7777 + 888 + 9999
25425 = 1 + 2 + 33 + 4 + 55 + 6666 + 7777 + 888 + 9999
25426 = 1111 + 22222 + 444 + 555 + 7 + 88 + 999

Q2. 

The minimum prime is 32027.

31991 = 22222 + 5 + 777 + 8888 + 99
32003 = 22222 + 444 + 555 + 6 + 7777 + 999
32009 = 22222 + 3 + 44 + 66 + 777 + 8888 + 9 

Q3.

The minimal number is already the smallest, so here are the first 3 pan digitals that can be made with repdigits.

123457896 = 11111 + 2222 + 33 + 444444 + 55555555 + 66666666 + 777777 + 88
123458967 =  11111 + 222222 + 3333 + 4 + 55555555 + 66666666 + 77 + 999999
123458976 =  11111 + 2 + 3333 + 444444 + 55555555 + 66666666 + 777777 + 88

***

Oscar wrote:

Below 111111, there are 149 numbers which can't be expressed as required; 19 of them are primes.

 

About Q1.

25424 = 2+33+4+55+6666+7777+888+9999

25425 = 111+22+3+4+5555+66+777+8888+9999

25426 = 11111+22+3333+66+7+888+9999

25427 -> no solution.

 

About Q2.

31991 = 1+22222+33+5+66+7777+888+999

32003 = 1+333+4444+555+6+7777+8888+9999

32009 = 11+2+333+4444+555+7777+8888+9999

32027 -> no solution.

 

About Q3.

Below 10^9, there are 9! = 362880 zero-free pandigitals, but only 14825 of them can be expressed as required.

In particular, there's no way to express the first nine of them: 

123456789, 123456798, 123456879, 123456897, 123456978, 123456987, 123457689, 123457698, 123457869.

These are the first three zero-free pandigitals for which there are solutions:

123457896 = 111111111+2222+33+44444+5555555+6666666+77777+88

123458967 = 111111111+22222+3333+4+5555555+6666666+77+99999

123458976 = 111111111+2+3333+44444+5555555+6666666+77777+88

***