20. Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied.
The digits used are not necessarily distinct.
ACROSS
1. Four less than a factor of 105 .
3. One more than a palindrome.
5. The square-root of the answer to this Kangaroo question.
DOWN
1. Two less than a square.
2. Four hundred less than a cube.
4. Six less than the sum of the answers to two of the other clues.
What is the square of the answer to 5 ACROSS?
SolUtion
841

The two-digit factors of 105 are 15,21 and 35 , so 1 ACROSS is one of 11,17 or 31 . However, as 2 DOWN is 400 less than a cube it cannot start with a 7 , so 1 ACROSS is either 11 or 31 .
If 1 ACROSS is 11 then 1 DOWN is 14,2 DOWN is 112 and 3 ACROSS is 415 . This means 4 DOWN is at least 50 which is too large to satisfy its clue, since 5 ACROSS is at most 29 .
So 1 ACROSS is 31 . This means 1 DOWN is 34,2 DOWN is 112 and 3 ACROSS is 415 . Checking each of the digits 1 to 9 in turn for the units digit of 4 DOWN yields 9 as the only possibility, with $59=31+34-6$.
Therefore, 5 ACROSS is 29 and its square is 841 .