Mark:
GRID
2316
and 9738
1296
3375
3136
1681
A start:
4 digit cubes
1000, 1331, 1728,
2197, 2744,
3375,
4096, 4913,
5832
6859
8000
9261
4 digit fourth powers
1296
2401
4096
6561
4 digit squares
1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936,
2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916,
3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969,
4096, 4225, 4356, 4489, 4624, 4761, 4900,
5041, 5184, 5329, 5476, 5625, 5776, 5929,
6084, 6241, 6400, 6561, 6724, 6889,
7056, 7225, 7396, 7569, 7744, 7921,
8100, 8281, 8464, 8649, 8836,
9025, 9216, 9409, 9604, 9801,
Firstly, since we know that 2 rows must contain a cube and a fourth power, as must 2 columns, we can try out some different combinations that match this rule.
GRID 1
1296
3375
3--6
1--1
GRID 2
--46
--05
1296
9261
GRID 3
6--4
5--0
6859
1296
GRID 4
9--4
2--0
6859
1296
GRID 5
9261
--82
--59
4096
GRID 6
6561
--82
--59
4096
GRID 7
1331
23--
97--
6561
GRID 8
--19
--22
4096
6561
So we now have 8 possible grids. However, look closely at them and you will see there are duplicates, just of a different orientation. I'll match them up now.
Grid 1 -> Grid 7
Grid 2 -> Grid 8
Grid 3 -> Grid 6
Grid 4 -> Grid 5
Which leaves us now with only 4 possible grids, which will be grids 1 to 4.
GRID 1
Possible squares to be entered are: 2304, 3136, 1521, 1681
We can discount 1521 because we know there must 4 odd and 4 even numbers, and so the middle 2 numbers in the fourth row must both be even. We try 2304 in column 2. However, with this, there is no other square number that will fit anywhere. So now we try 3136 in the third row, meaning 1681 is the only other possible square that fits in and it goes in the fourth row. And believe it or not, this solution actually works, so we don't need to bother checking the other grids!
Here is the final grid:
1296
3375
3136
1681
Lets check it meets all the requirements:
4 odd numbers: 3375, 1681, 1331, 6561
4 even numbers: 1296, 3136, 2316, 9738
Fourth power row: 1296
Fourth power column: 6561
Cube row: 3375
Cube column: 1331
Two square numbers: 3136, 1681
And so to answer the initial question:
Two numbers must not be perfect powers: 2316, 9738
A check on the other grids, just in case....
GRID 2
There are no square numbers ending in 19, 22, 46 or 05, therefore this grid will not work.
GRID 3 AND 4
A quick look at these and you can see that these already fail one of the requirements in the question. They both have 5 even numbers and only 3 odd numbers.
Which makes grid number 1 the definitive solution, even though I have no proof that 9738 and 2316 are not perfect powers.
TTH wrote,
"Looks more like you're drawing pictures"
I will have you know that is the latest design for my new Super Hero costume
'Duck Man'. He quacks jokes like:
How do you turn a duck into a soul-singer?
- Put it in a microwave until its bill withers.
Mis wrote, "
I have to remember just because they are hard to me doesn't mean it is for this group! Sheesh."
Even the simplest question becomes impossible if you don't know the answer! :wink:
Anyhoo, good to see you up and about on this fine brisk morning.
LCOMOBATCIONK
SFTEEASTK
Have a great Sunday - I am off to feed the young'uns! 
