As the Jeremy Lin sensation goes on, Roger, who is a geek and a super fan of Jeremy Lin, decides his new cell phone number must be "Linsane". More specifically, he wants his new phone number to satisfy:
Adjacent sum:
There is at least one occurrence in the phone number of three adjacent digits summing to x, where x is Lin's jersey number at New York Knicks.Diversity:
There are at least y different values of the digits used in the phone number, where y is Lin's jersey number at Golden State Warriors.Neighboring difference:
There is at least one pair of neighboring digits whose difference is equal to z, where z is Lin's jersey number at Harvard.
A phone number with length n contains n digits. Each digit is in the
range from 0 to 9, except that the first digit must be non-zero.
A phone number is called "linsane" if it satisfies the three constraints
listed above.
For phone numbers with a given length n, Roger wonders how many "linsane"
phone numbers exist.
He also wants to find out the "most linsane" phone number among them.
For a given length, the "most linsane" phone number is a "linsane" phone
number that has the biggest "linsanity measurement" among them.
"Linsanity measurement" is defined as , where n is the number of
digits and di is the i-th digit in the phone number.
If there is a tie on such measurement, choose the one whose median of the
digits is largest; and if there is still a tie, choose the largest phone
number.
Median is the (n+1)/2-th smallest digit if n is odd, or the average of
the (n/2)-th and (n/2+1)-th digit if n is even. For example, the
linsanity measurement of number 78969251 is equal to (159)%8 + (176)%8 +
(159)%8 + (152)%8 +(115)%8 + (71)%8 = 40 with its median equal to 6.5.
Input
The first line contains a positive integer T, the number of test cases. T test cases follow.
Each test case is a single line and contains exactly four integers separated
by single white space: n x y z, where n is the length of the phone
number, x is Lin's jersey number at New York Knicks, y is Lin's jersey
number at Golden State Warriors and z is Lin's jersey number at Harvard.
(x,y and z are not necessarily 17, 7 and 4 in another parallel
universe.)
Constraints
3 ≤ n ≤ 20
0 ≤ x ≤ 27
0 ≤ y ≤ 10
0 ≤ z ≤ 9
1 ≤ T ≤ 15
Among the T test cases, there will be no more than 5 test cases with n
Output
For each of the test cases numbered in order from 1 to T, output "Case #", followed by the case number, followed by ": ", followed by the number of possible "linsane" phone numbers mod 1018 for the given length for that case, and then a single space " " followed by the "most linsane" phone number for the given length or -1 if no "linsane" phone number exists for the given length.