SOLUTION: the digits 1,1,2,2,3 and 3 are arranged to form an odd six digit integer. the 1s are separated by one digit, the 2s by two digits, and the 3s by three digits. what are the last t

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 



:)