[Kamog]

actaully kamog, i would be interesting in knowing how you came up with that answer.

OK, this is how I got my (wrong) answer:

The question was:
????546456

Let's label the digits as follows:

code:

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?  ?  ?  ?  5  4  6  4  5  6

k0 k1 k2 k3 k4 k5 k6 k7 k8 k9

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I started off guessing that we are supposed to figure what the four numbers represented by ???? are, by looking at some sort of pattern in the Last six digits that are given. If this original premise is wrong, then my whole reasoning is wrong.

We can see that the next number in the series is not solely based on some sort of formula that uses the previous number, as k5 and k7 are both 4, but the next digits, k6 and k8, are not the same. Therefore, the next number may be based on two or more of the previous numbers using some sort of rule.

Furthermore, the next number is not based on the previous two numbers in the series. Notice that k4 = 5, k5 = 4, and k6 = 6. If k6 were based on some formula that uses only k4 and k5, then the rule may be commutative (meaning it gives the same result when the order of the two numbers is switched), as the same two numbers later produce the same result (k7 = 4, k8 = 5, k9 = 6).

However, this rule doesn't work elsewhere because k5 = 4, k6 = 6, which produces k7 = 4, but k6 = 6, k7 = 4, which results in k8 = 5, a different number. Therefore, the next number is not based just on the previous two numbers.

So I looked for patterns in the numbers.
(1) If you take one of the digits, skip the next digit, and then subtract it from the first number, you get the following pattern.

code:

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k4 - k6 = -1,  k5 - k7 = 0,  k6 - k8 = -1, k7 - k9 = -2

5  - 6  = -1,  4  - 4  = 0,  6  - 5  = -1, 4  - 6  = -2 

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code:

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Thus we have:          ?,  ?,  ?,  ?,  -1, 0, -1, -2.

So a pattern might be  -5, -4, -3, -2, -1, 0, -1, -2.

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Then it follows that k3 - 4 = -2, so k3 = 2; k2 - 5 = -3, so k2 = 2, k1 - 2 = -4, so k1 = -2, and k0 - 2 = -5 so k0 = -3.
Then the answer will be

code:

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-3 -2  2  2  5  4  6  4  5  6

k0 k1 k2 k3 k4 k5 k6 k7 k8 k9 

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However, I rejected this solution because of the negative numbers.

(2a) Notice that k4 + k5 = 9, k6 + k7 = 10, k8 + k9 = 11. We have an arithmetic progression where the sum of each pair of numbers increases by 1 for each successive pair. Continuing this pattern, we have k0 + k1 = 7 and k2 + k3 = 8.

(2b) Also, the odd pairs, k5 + k6 = 10, and k7 + k8 = 9. From this pattern, we have k1 + k2 = 12 and k2 + k3 = 11.

So we seem to have two progressions: the sum of the even pairs are increasing by 1 with each pair, and the sum of the odd pairs are decreasing by 1 with each pair.

Using this pattern, we get the following answer by solving (2a) and (2b) for k0, k1, k2, and k3.

code:

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-3 10  2  6  5  4  6  4  5  6

k0 k1 k2 k3 k4 k5 k6 k7 k8 k9

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This answer, too, was rejected because of the -3 and the 10. (We can't have two digits within one digit!)

(3) Notice that k9 + k7 = 10 and k7 + k5 = 8. If this pattern is carried back, we should have k5 + k3 = 6 and k3 + k1 = 4.
Then, 4 + k3 = 6, therefore k3 = 2, and from 2 + k1 = 4, we get k1 = 2.

If we solve for k0, k1, k2, k3 by using (2a) and (3) we get:

code:

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 5  2  6  2  5  4  6  4  5  6

k0 k1 k2 k3 k4 k5 k6 k7 k8 k9 

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Which seems like an OK answer

If we solve it using (2b) and (3) we get:

code:

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?   2 10  6  5  4  6  4  5  6

k0 k1 k2 k3 k4 k5 k6 k7 k8 k9

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Which is invalid because we don't know what k0 is, and k2 has two digits.

So now I stopped looking and submitted 5262546456 as my answer, which is the only 'good' answer I found.

But it turns out this is wrong and the correct answer is "654645????". I don't know why, only Sparhawk knows. Looks like he simply reversed the order of the string of characters.