# Magic: the Gathering is as Hard as Arithmetic

arXiv: https://arxiv.org/abs/2003.05119 · HTML: https://mlsanity.com/paper/2003.05119/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2020.

## Built on by (most cited first, 3 of 3)

- [Lessons from the Trenches on Reproducible Evaluation of Language Models](https://mlsanity.com/paper/2405.14782/) (2024)
- [What can we know about that which we cannot even imagine?](https://mlsanity.com/paper/2208.03886/) (2022)
- [A Taxonomy of Collectible Card Games from a Game-Playing AI Perspective](https://mlsanity.com/paper/2410.06299/) (2024)

## Abstract (arXiv)

Magic: the Gathering is a popular and famously complicated card game about magical combat. Recently, several authors including Chatterjee and Ibsen-Jensen (2016) and Churchill, Biderman, and Herrick (2019) have investigated the computational complexity of playing Magic optimally. In this paper we show that the ``mate-in-n'' problem for Magic is Delta0n-hard and that optimal play in two-player Magic is non-arithmetic in general. These results apply to how real Magic is played, can be achieved using standard-size tournament legal decks, and do not rely on stochasticity or hidden information. Our paper builds upon the construction that Churchill, Biderman, and Herrick (2019) used to show that this problem was at least as hard as the halting problem.