# Catch Up

## A simple yet compelling game

16 January 2021

Every so often a new mathematical game or puzzle is invented that is so simple that it seems incredible that nobody thought of it before. Think of Sudoku, which took the world by storm in the 1990s, or the matching game Dobble which became an international hit around 2010.

I wonder if the next blockbuster will be Catch-Up. This simple adding game was invented by a group of academics at New York University* in 2015, but has yet to catch on. The game can be played with stacks of Lego bricks of different heights, the aim being to end with a taller tower than your opponent. For example the 6-game is played with Lego stacks of height 1, 2, 3, 4, 5 and 6.

To start the game, Player A takes a single stack of any height to start their tower. Player B then takes stacks one at a time, building them up until their tower is the same height or taller than Player A’s, which ends their turn. Player A now does the same, stacking until their tower is at least as tall as Player B’s. The players keep taking turns until all of the stacks of bricks have been used up.

Here's an example for a game of six stacks (the N=6 game).

A went first with the 3 stack.

B responded with the 6 stack (end of turn).

A then played 2 followed by 5 (end of turn).

Finally B played 1 and then 4, to win by a margin of one. (Notice that B had to play 1,4 in that order, if they'd played 4 first then their tower would be the same height as A's which would be the end of B's turn.)

It's a very easy game to play - a six year old can understand it. Yet the tactics are far from obvious. In a game where N=10, a six year old could play an expert and still have a chance of winning. And the catch-up nature of the game means that whoever is behind after one move is by definition level or ahead after the next, so both players still feel like they are in with a chance until the final stages.

Catch Up has turned out to be a game of remarkable hidden depths. For some values of N, for example N=5 or N=6, it is known that Player A can force a win; for other values (eg N= 9 or 10) Player B can force a win. And in some games such as N=7 or (trivially) N=3, either player can force a draw with optimal play. But it's still not known if the game favours A or B overall, and the strategy for high values of N, eg N=50, is chess-like in its complexity, and it's still not known which player is favoured.

I predict that one day a games manufacturer will package this game in an appealing way (they'll call it SkyScrapers or some such) and it will be an international hit. But until they do, it can be your and my little secret that we play with friends or (in my case) with my children.

** My special thanks to Steve Brams and Mehmet Ismail, two of the game theorists who devised Catch Up, with whom I've had many interesting discussions about the game. You can find an introduction to their 2015 paper about Catch-Up here: http://gapdjournal.com/issues/issue-1-2/issue-1-2-sample-05-catch-up.pdf*