15. 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.
ACROSS
DOWN
1. A square
1. 4 down minus eleven
3. The answer to this
2. One less than a cube
Kangaroo question
4. The highest common factor
5. A square of 1 down and 4 down is greater than one
15. 829 The possible answers for 1 across are $16,25,36,49,64$ and 81 .
The possible answers for 2 down are 124, 215, 342, 511, 728, 999 .
By considering the last digit of 1 across (which must be the same as the first digit of 2 down) we see that the only possible pairs of answers for 1 across and 2 down are (25, $511),(49,999)$ and $(81,124)$. The pair $(49,999)$ would leave no possible answer for 5 across, so may be disregarded. The pair $(25,511)$ gives an answer of 16 for 5 across and thence 36 for 4 down and 25 for 1 down. However these answers contradict the clue for 4 down so this case may be disregarded. $(81,124)$ gives an answer of 49 for 5 across and hence 99 for 4 down and 88 for 1 down. These answers satisfy all the conditions in the clues and therefore the answer to this Kangaroo question is 829 .