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 From Minus One to Plus One (Posted on 2007-06-13)
``` +---+---+---+---+
|   |   |   |   |
+---+---+---+---+
|   |   |   |   |
+---+---+---+---+
|   |   |   |   |
+---+---+---+---+
|   |   |   |   |
+---+---+---+---+```

In each cell of the above matrix, place 1, -1 or 0 in such a way that each row and column has a different total.

The solution I have posted is not unique.

Can you make one more larger matrix with the same conditions?

The inspiration for this puzzle came from Anand Rao at Puzzleteasers.

 See The Solution Submitted by Josie Faulkner Rating: 4.4000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: computer solution | Comment 5 of 6 |
(In reply to computer solution by Charlie)

Why you can't have both +4 and -4.  Proof by contradiction:

First, you clearly can't have +4 as a row and -4 as a column as this would require one cell to have both a +1 and a -1.

So fill in one row of all +1's and a second of all -1's

The 4 column totals can now only be -2,-1,0,+1,+2 so at least one of the other rows must have three +1's and a 0 so as to sum to 3 (the missing sum would be -3 "to avoid trivial variations")

`+1 +1 +1 +1+1 +1 +1  0.. .. .. ..-1 -1 -1 -1`

The first 3 columns must be different so fill in a +1, 0, -1

`+1 +1 +1 +1+1 +1 +1  0+1  0 -1 ..-1 -1 -1 -1`

The last open space cannot be filled because its column and row total will always be the same.  Therefore you can't have sums of both +4 and -4

Edited on June 14, 2007, 9:06 am

Edited on June 14, 2007, 9:07 am
 Posted by Jer on 2007-06-14 09:05:22

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