Arranging Integer Numbers on a Loop Such That the Sum of any Two Adjacent Numbers Is a Perfect Square

This paper presents a systematic method (based on the concept of trios) for arranging integer numbers 1 to n on a loop such that the sum of adjacent numbers is a perfect square. The integer numbers are divided into two groups; group A consists of numbers ≤ 31 and rest in another group B. The numbers in group A do not form a closed loop; rather they form single or multiple open loops. For numbers in group B, there exists at least one closed loop that satisfies the perfect square condition. The number of loops increases at a very large rate with the increase in n and, therefore, cannot be handled manually. Therefore, an algorithm is developed. It is found that the total number of closed loops increases rapidly with increasing n above 43. A procedure for obtaining a closed loop of a higher number from that of a lower number is outlined. Finally, some anticipated applications of the theory are mentioned.