MathDB

3

Part of 2013 European Mathematical Cup

Problems(2)

Lock-breaking codes

Source: European Mathematical Cup 2013, Junior Division, P3

7/3/2014
We are given a combination lock consisting of 66 rotating discs. Each disc consists of digits 0,1,2,…,90, 1, 2,\ldots , 9 in that order (after digit 99 comes 00). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position 000000000000, and we know that this combination is not correct.
a) What is the least number of moves necessary to ensure that we have found the correct combination? b) What is the least number of moves necessary to ensure that we have found the correct combination, if we know that none of the combinations 000000,111111,222222,…,999999000000, 111111, 222222, \ldots , 999999 is correct?
Proposed by Ognjen Stipetić and Grgur Valentić
rotationinductiongeometrygeometric transformationcombinatorics proposedcombinatorics
Problem 3

Source: European Mathematical Cup

12/16/2013
We call a sequence of nn digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code (1101)(1101) contains 1010 palindromes, of which 66 are nice.)
a) What is the least number of palindromes in a code?
b) What is the least number of nice palindromes in a code?
combinatorics unsolvedcombinatorics