In the six rows of numbers below, each of the pairs adds up to 25. Now 25 happens to be a perfect square.

Fill in the blanks with a third number (a different number in each row) so that the sums of any two numbers on any row is a perfect square.

+---+---+---+
| 1 |24 |   |
+---+---+---+
| 2 |23 |   |
+---+---+---+
| 3 |22 |   |
+---+---+---+
| 4 |21 |   |
+---+---+---+
| 5 |20 |   |
+---+---+---+
| 6 |19 |   | 
+---+---+---+

DEFDBL A-Z
CLS
FOR n1 = 1 TO 10
n2 = 25 - n1
FOR n3 = 1 TO 999999
IF n3 <> n1 AND n3 <> n2 THEN
s2 = n1 + n3: sr = INT(SQR(s2) + .5)
IF sr * sr = s2 THEN
s3 = n2 + n3: sr = INT(SQR(s3) + .5)
IF sr * sr = s3 THEN
PRINT n1; n2; n3, s2; s3
EXIT FOR
END IF
END IF
END IF
NEXT
NEXT

finds

array      the 2 added squares
1 24 120    121 144
2 23 98     100 121
3 22 78     81 100
4 21 60     64 81
5 20 44     49 64
6 19 30     36 49

Also it can be extended two more rows:

1 24 120    121 144
2 23 2      4 25
3 22 78     81 100
4 21 60     64 81
5 20 44     49 64
6 19 30     36 49
7 18 18     25 36
8 17 8      16 25

 

by not requiring uniqueness of numbers within a row. I think that was a misunderstanding on my part that added that requirement.

Edited on June 10, 2013, 2:29 pm

Posted by Charlie on 2013-06-10 13:37:54