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 |   | 
+---+---+---+

Inspection of Charlie's solution leads to a nice pattern.

The numbers in the chart are n and (25-n)

If we add (9-n)(16-n)=n-25n+144 to each we get

n²-24n+144 = (n-12)² and n²-26n+169 = (n-13)²

The numbers are based on the 5,12,13 pythagorean triple.

Edited on June 10, 2013, 2:21 pm

Posted by Jer on 2013-06-10 14:15:21