Find The Code

First consider hints 1: $\boxed 4 \boxed 7 \boxed 1 \boxed 0 \boxed 9$ and hint 6: $\boxed 9 \boxed 0 \boxed 4 \boxed 3 \boxed 5$ both "One number is correct and correctly placed". Since $0$ , $4$ , and $9$ appear in both hints and none of them is at the same position in both hints, this means that they are all wrong numbers. Shade out the wrong numbers for better visual and we have the diagram on the left. Now we can clearly see the correct and correctly place number in hint 1 is either $7$ or $1$ and that of hint 6 is either $3$ or $5$ .

Second diagram: From hint 5, $\boxed 7 \boxed 5 \boxed 9 \boxed 6 \boxed 2$ , we know that the wrong numbers are continuously placed. Since $9$ is wrong, either $5$ or $6$ is wrong, but $7$ and $2$ are definitely correct. Let's highlight the correct number and may not be correctly placed with yellow.

Hint 1: $\boxed 4 \boxed 7 \boxed 1 \boxed 0 \boxed 9$ Now that we know that $7$ is correct and there is only one correct and correctly placed in this hint, $7$ must be the number. Let's highlight correct and correctly placed number with green. Then $1$ in this hint must be wrong. Shade out $1$ in all the hints.

Hint 4: $\boxed 2 \boxed 4 \boxed 6 \boxed 9 \boxed 1$ After shading out $1$ , the remaining $2$ and $6$ must be correct as mentioned by this hint.

Hint 5: $\boxed 7 \boxed 5 \boxed 9 \boxed 6 \boxed 2$ Now $5$ must be the wrong number.

Hint 6: $\boxed 9 \boxed 0 \boxed 4 \boxed 3 \boxed 5$ What remains in the hint is $3$ which is correct and correctly placed.

hint 2: $\boxed 8 \boxed 4 \boxed 5 \boxed 7 \boxed 0$ What remains in the hint are $8$ and $7$ . $8$ must be the other correct number but wrongly placed.

Now we have the five correctly number $2$ , $3$ , $6$ , $7$ , and $8$ , with two numbers correctly place, $3$ (fourth place) and $7$ (second place). From hints 2 and 3, we know that $2$ and $8$ cannot be at the first place $6$ must be at the first place. From hint 4, $2$ and $6$ must be separated by a number, $2$ must be in the middle, leaving $8$ to the last. Therefore the code is $\boxed 6 \boxed 7 \boxed 2 \boxed 3 \boxed 8$ .

$\text{First let us number the hints from top to bottom as Hint }1\text{ to Hint }6$

$\text{Taking the Hint }5\text{ first : possibilities of wrong numbers are }\newline (a)\, 7\text{ and }5\text{ or}\newline(b)\,5\text{ and }9\text{ or}\newline(c)\,9\text{ and }6\text{ or}\newline(d)\,6\text{ and }2$

$(a) \text{ cannot be true because if }7\text{ and }5\text{ are wrong numbers that means }9,6\text{ and }2\text{ are correct numbers which makes the statement of Hint 4 false.}\newline (c) \text{ cannot be true because if }9\text{ and }6 \text{ are wrong numbers that means }7,5\text{ and }2\text{ are correct numbers which makes the statement of Hint 3 false.}\newline (d)\text{ cannot be true because if }6\text{ and }2\text{ are wrong numbers that means }7,5\text{ and }9\text{ are correct numbers which makes the statement of Hint 1 false.}\newline (b) \text{ is true because no other statements are contradicting this. Hence }5\text{ and }9\text{ are wrong numbers and }7,6\text{ and }2\text{ are correct numbers}\newline \text{Taking this information and going to Hint }3\text{ we will get that }0,4\text{ and }1\text{ are also wrong numbers. So till now }0,1,4,5\text{ and }9\text{ are wrong numbers.}\newline \text{In Hint }2\text{ we know }0,4\text{ and }5\text{ are wrong numbers which means }7\text{ and }8\text{ are correct numbers.}\newline \text{In Hint }6\text{ we know }0,4,5\text{ and }9\text{ are wrong numbers which means }3\text{ is the correct number. So we get }2,3,6,7\text{ and }8\text{ are correct numbers.} \newline \text{Positions of }7\text{ and }3\text{ are known using Hint 1 and Hint 6 respectively. Now remaining vacant positions are Position 1,3 and 5. Number } 8 \text{ cannot }\newline\text{be at Position 1 because of Hint 2 and also not at Position 3 because if it is placed at position 3 then 2 and 6 will have 3 numbers between them.}\newline\text{Hence 8 will be placed at position 5. Now 2 cannot be at Position 1 because of Hint 3 so 2 will be at position 3 and 6 will be at position 1} \newline$

$\text{Hence the code is } \boxed{6}\boxed{7}\boxed{2}\boxed{3}\boxed{8}. \text{ So the answer is }67238.$