## 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.

##### More Answers:

RSA EncryptionMaximum Product of Parts

Triangles Containing the Origin