Let the 6 digits of the number be D1 through D6, with D6 being the unit's place.



Since it is an odd number, the last digit has to be only 1 or 3.



Can the last digit be 1? Let us check logically.



If the last digit (D6) has to be 1, then D4 has to be 1 too (since

1's are separated by 1 digit).

Now, D4 and D6 are taken.

If the 3's have to be separated by 3 digits, they can now only occupy D1 and D5.

That leaves only D2 and D3 vacant. But if we put the 2 2's there, they will not

be separated by 2 digits.



Hence, the last digit CANNOT be 1 and has to be only 3.



If D6 is 3, then D2 has to be 3.

If the two 2's are to be separated by 2 digits, they can now occupy D1 and D4.

(The other possible combinations which are separated by 2 digits are D2-D5 and

D3-D6 which are not possible, since D2 and D6 are taken)

That leaves the two 1's in D3 and D5 which are separated by 1 digit.



So the number is  and the last 3 digits are 



:) 

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