20 4 16 37 58 89 42. This sequence doesn't make sense. However, if you add a number among them, it becomes a cyclical sequence. Which number do you need to add, and between which numbers?

This numerical puzzle reveals a hidden pattern once a crucial missing piece is identified. The rule governing this sequence is to take each digit of a number, square it, and then add those squared digits together to produce the next number in the series. Let's trace the given numbers: starting with 20, we square its digits (2^2 + 0^2) to get 4. Applying the same rule to 4 yields 4^2, which is 16. Continuing this process, 16 transforms into 1^2 + 6^2 = 37, and 37 becomes 3^2 + 7^2 = 58. Finally, 58 leads to 5^2 + 8^2 = 89.

The sequence as provided initially breaks after 89. To complete the cyclical pattern, the number 145 must be inserted between 89 and 42. When we apply the rule to 89, we find that 8^2 + 9^2 equals 64 + 81, which is indeed 145. Subsequently, applying the rule to this newly found number, 145, we calculate 1^2 + 4^2 + 5^2, which sums to 1 + 16 + 25, resulting in 42. This calculation successfully closes the loop, as 42 is already present in the original sequence, making it a perfectly cyclical progression.

This intriguing type of number sequence is a classic example found in recreational mathematics, often associated with the concept of "happy numbers" and "unhappy numbers." While happy numbers eventually reduce to 1 through this iterative squaring and summing of digits, unhappy numbers, like those in this puzzle, will inevitably fall into a repeating cycle. The specific eight-number cycle demonstrated here (20, 4, 16, 37, 58, 89, 145, 42) is a well-known mathematical curiosity, representing the only cycle that unhappy numbers enter, a testament to the surprising patterns hidden within basic arithmetic.