Please restore the 5 by 5 grid containing 10 ternary numbers with no leading zeroes.
The crossword-like definitions follow:


Across:
I. a cube
II. twice a permutable (in base 10) prime
III. divisible by eleven
IV. the number of trees with 10 vertices
V. a square

Down:
1. repdigit in base 12
2. a factorion in base 10
3. a semiprime
4. a square of a prime
5. its cube in base 10 uses only 3 distinct digits

Rem: I have built it bottoms-up. Hope there is only one solution...

The condition that there were no leading zeroes means the solutions ranged from 81 (10000 in base 3) to 242 (22222 in base 3), and in particular that 1 across and 1 down don't contain any zeroes.

The number of trees with 10 vertices is 106, and the only valid base 10 factorion is 145, so 4 across and 2 down could be filled in immediately with (10221) and (12101), respectively.

1 across: The only cubes in the range of solutions are 125 (11122) and 216 (22000) but since this number can't contain zeroes it must be 125 (11122).

4 down: The only squares of primes in this range are 121 (11111) and 169 (20021), and we know that 4 down starts with a 2, so it must be 169 (20021).

1 down: The only solutions in the range that are both repdigit in base 12 and don't contain any zeroes in base 3 are 130 (11211) and 157 (12211).  So we don't know what the second digit is, but we can fill in the rest (1x211).

5 across: We know that it's 11x1x.  The only square that fulfills that condition is 121 (11111).

5 down: We know that it's 2xx11 and that it's second digit is either 1 or 2.  There are only six possible solutions, and only one of them matches the clue, 211 (21211).

2 across: We know that it's x2x01.  The only values in the range that are double a permutable prime are 146 (12102) and 226 (22101).  It must be 226 (22101).

3 across: We know it's 21x02.  There are only three possibilities, and only one of them is divisible by 11, it's 209 (21202).

The final solution is:

1 1 1 2 2

2 2 1 0 1

2 1 2 0 2

1 0 2 2 1

1 1 1 1 1

Edited on March 29, 2014, 8:11 am

Posted by tomarken on 2014-03-28 14:08:28