Place the numbers 1 through 15 in a 3 by 5 array so that each column has the same sum and each row has the same sum.

Every row must have the same sum equal to 40 and every column must have the sum equal to 24. Now there are 8 odds and 7 evens from 1 to 15.


No row can have even number of even number because if it is true then it has odd number of odd numbers. So number of evens in a row can be only 1,3 or 5.

If a row has 5 evens then the other two rows must have 1 even number or of the form of (5,1,1). Similalry no row has a 5 even numbers then we have (3,3,1) as the second combination.

Similarly we can workout for columns. Every column can have either 1 or 3 even numbers(because the sum is even.


Now one of possible forms of our answer is given below:



E  E  E  O  O

E  O  O  E  E

E  O  O  O  O


where E means Even and O means Odd.



Now proceeding by trial and error I got to one of the solutions.



12  8  4  3  13

10  1  9  14  6

2  15  11  7  5

Posted by Sandeep on 2004-12-09 16:44:20