Endless love: On the termination of a playground number game

Abstract

A simple and popular childhood game, `LOVES' or the `Love Calculator', involves an iterated rule applied to a string of digits and gives rise to surprisingly rich behaviour. Traditionally, players' names are used to set the initial conditions for an instance of the game: its behaviour for an exhaustive set of pairings of popular UK childrens' names, and for more general initial conditions, is examined. Convergence to a fixed outcome (the desired result) is not guaranteed, even for some plausible first name pairings. No pairs of top-50 common first names exhibit non-convergence, suggesting that it is rare in the playground; however, including surnames makes non-convergence more likely due to higher letter counts (for example, `Reese Witherspoon LOVES Calvin Harris'). Different game keywords (including from different languages) are also considered. An estimate for non-convergence propensity is derived: if the sum $m$ of digits in a string of length $w$ obeys $m > 18/(3/2)^{w-4}$, convergence is less likely. Pairs of top UK names with pairs of `O's and several `L's (for example, Chloe and Joseph, or Brooke and Scarlett) often attain high scores. When considering individual names playing with a range of partners, those with no `LOVES' letters score lowest, and names with intermediate (not simply the highest) letter counts often perform best, with Connor and Evie averaging the highest scores when played with other UK top names.