|
%I #18 Jul 12 2017 15:19:57
%S 1,12,124,1251,12721,169896,8985898
%N Lexically smallest string of n digits from 1...9, such that no formula using the single digits in the given order exists that evaluates to 0.
%C The formula may use any combination of the four binary operators +, -, *, /, the unary minus and parentheses. All digits have to be used exactly once and in the given order. Concatenation of the digits is not allowed. 0/0 in the evaluation of expressions makes the result invalid.
%C The last term a(7)=8985898, which is the only string of 7 digits for which it is not possible to construct an expression with result 0, is conjectured to be the last sequence term, i.e. for all strings of 8 or more digits expressions with result 0 can be found.
%C The conjecture is almost trivially true. Since 8985898 is the only exception for strings of 7 digits, an 8 digit string containing 8985898 can only be of the form x8985898 or 8985898x. But since any of x898589 and 985898x /= 8985898 leads to a 7 digit string that can represent 0, one can always find a substring of length 7 representing 0 in a string of 8 or more digits. Therefore the sequence ends at a(7).
%H IBM Research, <a href="https://www.research.ibm.com/haifa/ponderthis/challenges/June2017.html">Ponder This Challenge - June 2017</a>, related problem for 7-digit numbers.
%e a(3)=124: For all numbers starting with 11.. the formula can be chosen as (1-1)*..,
%e 121: 1-2+1=0, 122: (1-2/2)=0, 123: 1+2-3=0, 124: 0 cannot by obtained by any valid formula.
%Y Cf. A000108 (number of ways to insert the parentheses), A181898, A288351, A288353, A288354, A288355, A288356.
%K nonn,base,fini,full
%O 1,2
%A _Hugo Pfoertner_, Jun 08 2017
|
