20. Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.
What is the answer to 3 ACROSS?
ACROSS
1. A multiple of 7
3. The answer to this Question
5. More than 10
DOWN
1. A multiple of a square of an odd prime; neither a square nor a cube
2. The internal angle of a regular polygon; the exterior angle is between $10^{\circ}$ and $20^{\circ}$
4. A proper factor of 5 ACROSS but not a proper factor of 1 DOWN
Solution
961
2 DOWN has possible answers of 162,165 and 168 .
1 ACROSS must end with a ' 1 ' so can only be 21 or 91 .
1 DOWN must start with a ' 2 ' or a '9'. The squares of odd primes are $9,25,49$. The only multiples of these which are neither square nor cube and which start '2' or ' 9 ' are 90 (which cannot be correct since 3 ACROSS cannot start with a 0 ), 98 and 99. Hence 1 ACROSS is 91 . 5 ACROSS could start with a ' 2 ', '5' or ' 8 '.
However, the first digit cannot be ' 2 ' since none of $21 \ldots 29$ (for 4 DOWN) has a proper factor which is two digits long and shares the same units digit.
Similarly, the first digit cannot be ' 5 ' since none of $51 \ldots 59$ (for 4 DOWN) has a proper factor which is two digits long and shares the same units digit. Hence 2 DOWN is 168.
Of $81 \ldots 89$ the only number to have a proper factor which is two digits long and shares the same units digit is 84 with the factor 14. So 5 ACROSS is 84 and 4 DOWN is 14.
For 1 DOWN we now know that 14 cannot be a factor, so this is 99 . Therefore the answer to 3 ACROSS is 961 .