Why the Cycle Is Perfect

The ones digits of the 7s hit every digit 0 through 9 exactly once before they repeat. That is not luck. It happens because 7 and 10 are coprime — they share no common factor — so the loop cannot close early.

This hands you a free anchor in the middle. 7 × 7 = 49 is a perfect square and sticks easily. From there, 7 × 8 is just 49 + 7 = 56, and 7 × 6 is 49 − 7 = 42. One square, two neighbors.