I have put eight four-digit numbers together in a 4x4 grid, such that four can be read across and four downwards.

Four of my numbers are odd and four are even. I have one cubed number going across and one going down, and I have one fourth-power number going across and one going down. Two of the other numbers in the grid are squares.

Which two numbers in my grid are not perfect powers?

I looked at four-digit 4 powers first and found:
1296
2401
4096
6561

Placing 2401 or 4096 in the top row of grid will produce a 4-digit number of 0XXX which I assume doesn't count as a 4-digit number.

I placed 1296 in the top row of grid and then looked which of other 4th powers could fit in a column. 1st I tried 2401 in 2nd column but I couldn't find 3rd powers which would then fit in both a row and column. I then tried 6561 in 4th column then tried to find 3rd powers which would fit in a row and column and found 1331 in column 1 and 3375 in row 2 would work. I then looked for square numbers that would fit in columns 2 and 3 that were even numbers and found 2304 for column 2 but none beginning with 97. I then looked for even numbers to fit in rows 3 and 4 and found 3136 for row 3 and 1521 or 1681 for row 4. As there were 4 even and 4 odd the middle two digits for row 4 had to be even so I used 1681 and came up with the following grid:

1296
3375
3136
1681

4th powered: 1296 & 6561
cubed: 1331 & 3375
squared: 1681 & 3136
not perfect power: 2316 & 9738

This grid can also be written as follows:

1331
2316
9738
6561

Edited on July 11, 2005, 9:40 am

Posted by Lisa on 2005-07-11 09:37:53