# The couple of Bethlehem

Bethlehem invited seventeen friends to his birthday party. Each one was assigned a number from 2 to 18, with 1 being reserved for her. When everyone was dancing, he realized that the sum of each couple's numbers was a perfect square.

What is the number of the couple in Bethlehem?

#### Solution

The highest number that can be reached by adding those that Bethlehem has distributed is 35 = 17 + 18, and the lowest number 3 = 1 + 2. The squares that can be achieved are therefore: 4, 9, 16 and 25.

With this information, it is clear that the couple of Bethlehem can be 3, 8 or 15. To find out, we must place the other couples to know if there is a single solution, or there can be several situations.

Let's start by eliminating the easiest numbers. Since 18 is greater than 16, he can only be dancing with 7, to add 25. The same happens with 17, who has to dance with 8, and with 16, who dances with 9. Since 7, 8 and 9 are already paired , we suppress them and we will write down the other possibilities.

The 15 can be with 1 and with 10. The 14, with 2 and with 11. The 13, with 3 and with 12. The 12, with 13 and with 4. The 11, with 5 and with 14. The 10, with 6 and 15. 15. 6, 10 and 3. 3. 5, 11 and 4. 4. 4, 12 and 5. 3, 13, 6 and 1. And 2 can only dance with 14. With this information we assemble the following table that will help us rule out cases:

 Invited Possible couple 15 1, 10 14 2, 11 13 3, 12 12 4, 13 11 5, 14 10 6, 15 6 10, 3 5 11, 4 4 12, 5 3 13, 6, 1 2 14 1 15, 8, 3

Since 2 can only dance with 14, he is caught. The 11 is forced to dance with the 5. For this reason, the 4 mandatory dance with the 12. In turn, that forces the 13 to dance with the 3.

At this point, 6 has to dance with 10 and therefore 15 will dance with the 1 that Bethlehem is wearing.