Problems & Puzzles: Puzzles

 Puzzle 1030. A puzzle cousin of Puzzle 1027 In this puzzle Rodolfo Kurchan now asks for numbers that are equal to the sum of K<=10 integers A, B, ... such that the conjunction of these summands shown no repeated digits. Examples: 393 = 352 + 41 856 = 856 1000 = 987 + 13 1111 = 987 + 124  45 = 45 o 42 + 3 o 1+2+3+4+5+6+7+8+9 10002 = 9874 + 125 + 3 Q1. Find the smallest integer N that can not be obtained such way? Q2. Redo Q1 for N=prime.

During the week 20-26 February, 2021, contributions to this puzzle came from Oscar Volpatti, Emmanuel Vantieghem and Paul Cleary

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Oscar wrote:

I wasn't 100% sure about using digit "0" within a sum, as none of the given example sums involves it.
On the other side, a permitted sum of K=10 integers with no repeated digits is mentioned, but the only such sum is 0+1+2+3+4+5+6+7+8+9.
However, my program could handle different versions of the problem:
digits 0-9 are used at most once (standard?)
digits 1-9 are used at most once, digit 0 is forbidden (harder)
digits 1-9 are used at most once, digit 0 can be repeated (easier)
In each case, I searched up to 10^5, finding both prime and composite numbers with no representations.

Hard version: 16414
composite exceptions, 1741 prime exceptions.
Q1:   N = 6088
Q2:   P = 10889

Standard version: 756 composite exceptions, 51 prime exceptions (see attached file P1030.txt).

Q1:   6088 = 6087+1,   N = 18872
Q2:   10889 = 10857+32,   P = 22111

Easy version: 162 composite exceptions, 6 prime exceptions.
Q1:   18872 = 18570+302,   N = 22888
Q2:   22111 = 21807+304    P = 78787

As for puzzle 1027, some solutions for numbers before the ones without representations (standard version only)
18868 = 18496+372
18869 = 18596+273
18870 = 18596+274
18871 = 18496+375

22079 = 21695+384
22091 = 21786+305
22093 = 21789+304
22109 = 21769+340

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Emmanuel wrote:

The smallest non-representable number I could find is  18872.
For the three previous numbers I found :
18871 = 10375 + 8496
18870 = 10274 + 8596
18869 = 10273 + 8593
The smallest non-representable prime I could find  is  22111.
For the three previous primes I found :
22109 = 15230 + 6879
22093 = 15304 + 6789

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Paul wrote:

I cant find any solutions for Q1 at 18872 and Q2 at 105107 (wrong).

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