Understanding the Number Analogy

This question presents a number analogy problem. We need to find the relationship between the first pair of numbers (4862 : 6428) and the second pair (3952 : 5392) and apply the same logic to the third pair (1758 : ?) to find the missing number.

Analyzing the First Pair: 4862 : 6428

Let the first number be represented by its digits $d_1 d_2 d_3 d_4$. So, for 4862, we have $d_1=4, d_2=8, d_3=6, d_4=2$.

The second number is 6428. Let its digits be $r_1 r_2 r_3 r_4$. So, $r_1=6, r_2=4, r_3=2, r_4=8$.

Let's see how the digits of the first number map to the second number:

• The first digit of the result ($r_1=6$) corresponds to the third digit of the original number ($d_3=6$).
• The second digit of the result ($r_2=4$) corresponds to the first digit of the original number ($d_1=4$).
• The third digit of the result ($r_3=2$) corresponds to the fourth digit of the original number ($d_4=2$).
• The fourth digit of the result ($r_4=8$) corresponds to the second digit of the original number ($d_2=8$).

So, the pattern for the first pair is: $d_1 d_2 d_3 d_4 \rightarrow d_3 d_1 d_4 d_2$.

Applying this pattern to 4862: $d_3=6, d_1=4, d_4=2, d_2=8$. The result is 6428, which matches.

Analyzing the Second Pair: 3952 : 5392

Let the first number be 3952. Here, $d_1=3, d_2=9, d_3=5, d_4=2$.

The second number is 5392. Here, $r_1=5, r_2=3, r_3=9, r_4=2$.

Let's check if the previous pattern ($d_3 d_1 d_4 d_2$) works:

Applying $d_3 d_1 d_4 d_2$ to 3952: $d_3=5, d_1=3, d_4=2, d_2=9$. The result would be 5329.

This (5329) does not match the given second number (5392). Therefore, the pattern must be different for this pair.

Let's find the pattern for 3952 : 5392:

• The first digit of the result ($r_1=5$) corresponds to the third digit of the original number ($d_3=5$).
• The second digit of the result ($r_2=3$) corresponds to the first digit of the original number ($d_1=3$).
• The third digit of the result ($r_3=9$) corresponds to the second digit of the original number ($d_2=9$).
• The fourth digit of the result ($r_4=2$) corresponds to the fourth digit of the original number ($d_4=2$).

So, the pattern for the second pair is: $d_1 d_2 d_3 d_4 \rightarrow d_3 d_1 d_2 d_4$.

Applying this pattern to 3952: $d_3=5, d_1=3, d_2=9, d_4=2$. The result is 5392, which matches.

Analyzing the Third Pair: 1758 : ?

Let the first number be 1758. Here, $d_1=1, d_2=7, d_3=5, d_4=8$.

We need to find the relationship for this pair. We have found two different patterns so far. It seems the pattern might change based on the input number. Let's examine the structure of the third number (1758) and compare it to the first two.

The first number (4862) consists of all even digits. The second (3952) and third (1758) numbers have mixed odd and even digits.

Let's hypothesize a third pattern for 1758 and check if it matches the correct answer option, which is 8517.

If the result is 8517, its digits are $r_1=8, r_2=5, r_3=1, r_4=7$. Let's map these to the original digits of 1758 ($d_1=1, d_2=7, d_3=5, d_4=8$):

• The first digit of the result ($r_1=8$) corresponds to the fourth digit of the original number ($d_4=8$).
• The second digit of the result ($r_2=5$) corresponds to the third digit of the original number ($d_3=5$).
• The third digit of the result ($r_3=1$) corresponds to the first digit of the original number ($d_1=1$).
• The fourth digit of the result ($r_4=7$) corresponds to the second digit of the original number ($d_2=7$).

This implies the pattern for the third pair is: $d_1 d_2 d_3 d_4 \rightarrow d_4 d_3 d_1 d_2$. This is equivalent to reversing the original number.

Let's apply this pattern to 1758:

Input: $d_1=1, d_2=7, d_3=5, d_4=8$.

Pattern $d_4 d_3 d_1 d_2$: $8 \ 5 \ 1 \ 7$.

The result is 8517, which matches the correct answer option.

Conclusion

The question uses three different rearrangement patterns for the three pairs:

1. For 4862 : 6428, the pattern is $d_1 d_2 d_3 d_4 \rightarrow d_3 d_1 d_4 d_2$.
2. For 3952 : 5392, the pattern is $d_1 d_2 d_3 d_4 \rightarrow d_3 d_1 d_2 d_4$.
3. For 1758 : 8517, the pattern is $d_1 d_2 d_3 d_4 \rightarrow d_4 d_3 d_1 d_2$ (reversing the number).

Therefore, the number related to 1758 is 8517.