### Endless love: On the termination of a playground number game

- A well-known playground game aims to compute a "love score" between two players based on the letters in their names and demonstrates surprisingly rich mathematical behavior; we use some back-of-the-envelope maths and computer simulation to find the highest-scoring names, situations in which the game never ends, and general rules underlying its behaviour.

The "Love Calculator" game is played in playgrounds and online across the world. It computes a playful "love compatibility" by first writing down the counts of l's, o's, v's, e's and s's that appear in two partners' names, then repeatedly adding numbers written next to each other until a percentage score is found -- "Alice loves Bob 54%!" (11010 -> 2111 -> 322 -> 54, as in the picture).

Our light-hearted research project has used maths and computer simulation to explore how the game behaves with different names -- including pairs of all the most common childrens' names in the UK -- and in different languages. "Endless love" -- when the game gets stuck in a loop, or keeps expanding forever -- often occurs between names with high letter counts (like Reese Witherspoon and Calvin Harris).

The project found that any point in the game can be described as a point in a mathematical "space", and that each step in the game moves a point in different ways, like different pieces on a chessboard. While individual outcomes are hard to predict, the average behaviour shows patterns that are repeated over games. The space contains a "cliff"; if moves carry a game over the cliff, it will continue forever.

(left) Alice and Bob playing the "loves" game. The number of l's, o's, v's, e's, and s's in their names give the first string of numbers. Adding neighbouring numbers produces the next strings, until we arrive at the 54% score. But the game never stops for some name combinations. (right) A mathematical picture of the game. w is the length of a string of numbers; m is the sum of the numbers in the string (so 11010 would have w = 5 and m = 3). The arrows show how w and m change on average as a game progresses. The blue region moves left toward an m = 2 final score; the red region keeps growing (or looping), never reaching a final score, and leading to "endless love".

Different patterns of the "loves" letters give different expected scores. Among the most common childrens' names in the UK, Connor has the highest expected score of 67%, with Evie, Holly, Lola, Molly and Olivia also scoring highly. Names with no "loves" letters -- from Adam to Ryan -- have the lowest expected score of 26%. The most successful names have a middling number of "loves" letters, between 2 and 6. Pairs of o's, several l's, and an absence of some other letters seem to be the key to success -- though a full theory of which patterns give which scores remains elusive.

"Endless love: On the termination of a playground number game" is due to appear in Recreational Mathematics Magazine and is available here. Iain

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