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Square Power (Posted on 2005-07-10) Difficulty: 4 of 5
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?

No Solution Yet Submitted by Sam    
Rating: 4.5000 (2 votes)

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Solution by Hand | Comment 4 of 7 |

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

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