Problem 674

Solution
D
When you are faced with a crossnumber, the best strategy is to look for clues where it is easy to find a unique solution.
Among the three-digit integers there are more squares and cubes than fourth and fifth powers. So the best strategy is to begin with 1 Down and 3 Across.

The first few numbers that are twice fifth powers are $2 \times 1^{5}=2 \times 1=2$, $2 \times 2^{5}=2 \times 32=64,2 \times 3^{5}=2 \times 243=486$ and $2 \times 4^{5}=2 \times 1024=2048$.
We can deduce from this that 486 which is the only three-digit number in this list is the answer for 1 Down.

It follows that 3 Across is a three-digit fourth power with 6 as its hundreds digit. We have $3^{4}=81,4^{4}=256,5^{4}=625$ and $6^{4}=1296$.
We deduce that 3 Across is 625 .
We now see that 2 Down is three-digit cube with units digit 5 . Hence 2 Down is is $5^{3}=125$.
Finally, 1 Across is a three-digit square with hundreds digit 4 and units digit 1. Therefore 1 Across is $21^{2}=441$.

We can now deduce that $*$ is 4 .