20. Each cell in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.
What is the answer to 2 DOWN?

ACROSS
1. A prime which is the sum of two squares
3. Twice the answer to 2 DOWN
DOWN
1. $p \times q$, where $p, q$ are prime
and $q=p+4$
4. $60 \%$ of 5 ACROSS
Solution
397

The only two-digit $p q$ with $q=p+4$ are $77=7 \times 11$ and $21=3 \times 7$. The latter of these is not possible, since 3 ACROSS cannot begin with a 1 as it is twice a three-digit number.
So the answer to 1 DOWN is 77 .
We now know the first digit of 1 ACROSS is 7 , meaning that as primes 71,73 and 79 are possible. Of these, the only one that can be expressed as the sum of two squares is $73=64+9$. The clue for 4 DOWN tells us that $5 \times 4$ DOWN $=3 \times 5$ ACROSS. Hence 5 ACROSS is a multiple of 5 and 4 DOWN is a multiple of 3 . The first of these shows that the bottom right entry is 5 (since 0 is not allowed); and the second then shows that 4 DOWN is a multiple of 15 . By the clue for 3 ACROSS, the first digit of 4 DOWN is even. It cannot be $0,2,6$, or 8 because 4 DOWN is a multiple of 3 . Therefore 4 DOWN $=45$ and 5 ACROSS $=75$.
Finally, given the first digits of 2 DOWN and 3 ACROSS and the clue for 3 ACROSS, we see that the middle digit must be at least 5 and we then can check quickly that 9 is the only possible value.
Hence 397 is the solution.