Number Mind

The game Number Mind is a variant of the well known game Master Mind.
Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you’re only told in how many places you’ve guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you’d be told that you have one correct digit; however, you would NOT be told that you also have another digit in the wrong place.
For instance, given the following guesses for a 5-digit secret sequence,
90342 ;2 correct
70794 ;0 correct
39458 ;2 correct
34109 ;1 correct
51545 ;2 correct
12531 ;1 correct
The correct sequence 39542 is unique.
Based on the following guesses,
5616185650518293 ;2 correct
3847439647293047 ;1 correct
5855462940810587 ;3 correct
9742855507068353 ;3 correct
4296849643607543 ;3 correct
3174248439465858 ;1 correct
4513559094146117 ;2 correct
7890971548908067 ;3 correct
8157356344118483 ;1 correct
2615250744386899 ;2 correct
8690095851526254 ;3 correct
6375711915077050 ;1 correct
6913859173121360 ;1 correct
6442889055042768 ;2 correct
2321386104303845 ;0 correct
2326509471271448 ;2 correct
5251583379644322 ;2 correct
1748270476758276 ;3 correct
4895722652190306 ;1 correct
3041631117224635 ;3 correct
1841236454324589 ;3 correct
2659862637316867 ;2 correct
Find the unique 16-digit secret sequence.

The game Number Mind is a variant of the well known game Master Mind. The objective is to guess a sequence. Instead of provided with whether a number is correct or out of place, you are only given how many numbers you have guessed correctly. It’s like playing Master Mind but with fewer clues.

In order to solve this problem, we create a grid of possibilities.

We list all the numbers vertically in order of their appearance. Then we go clue by clue and use process of elimination. The guesses with only 1 correct number especially allows us to eliminate a lot of possiblities:

Clues:
1) The first digit cannot be 5 (15th line), it must be 1.
2) The second number cannot be 6 (15th line) or 8 (6th line), it must be 9.
3) The third digit can’t be 1 (6th line), 6 (15th line), or 5 (5th line), it must be 8.
4) The fourth digit can’t be 5 (5th line) or 1 (1st line), it must be 3.
5) Fifth digit can’t be 4 or 6 (2nd line), it must be 5.
6) The sixth digit can’t be 9 or 4 (2nd line), it must be 7.
7) Taking a look at the lines where there is a single correct figure, the seventh digit can’t be 3 (6th line), 0 (1st line), 8 (5th line), nor 1 (15th line), so, it’s a 2.
8) The eight digit can’t be 9 or 4 (2nd line), it must be 4.
9) The ninth digit can’t be 5 (7th line) or 6 (5th line), it must be 9.
10) The tenth digit can’t be 5 (7th line) or 1 (2nd line), it must be 6.
11) The eleventh digit can’t be 1 (2nd line), it must be 7.
12) The twelfth digit can’t be 8 (6th line), it can be 0.
13) The thirteenth digit can’t be 8 (6th line) or 5 (5th line), it must be 3.
14) The fourteenth digit can’t be 5 (5th line) or 9 (6th line), it must be 1.
15) The fifteenth digit can’t be 2 (2nd line), it can be 5.
16) Looking at searches #7 and #11, we can see only the numbers 4 and 7 are filled, therefore, the last digit is 4.

Therefore, the secret number is 1983572949670315414.

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