# How useful is Marijn's Bluff?

Parcly and Tori Taxel, after having wished genies' chess into existence and played around with it – noticing the link to Zarankiewicz's problem and getting an OEIS entry published in the process – decided to play Magic: The Gathering afterwards. While teaching each other the rules, however, they reached a position where Tori had exactly 161 cards on her side of the battlefield.

It was now Parcly's turn, and she played a card called Marijn's Bluff*, which read

Your opponent indexes the cards in their side of the battlefield starting from 1 as they like. You win the game if in this ordering there are $$x$$ and $$y$$ such that cards $$x,y,x+y$$ all exist and are all the same colour; $$x$$ may equal $$y$$ and multi-colour/colourless cards may be taken as any single colour.

Without resorting to other MtG cards/rules, and bearing in mind that the game has five colours, must Tori lose no matter what her 161 cards are?

*Obviously Parcly had magically written her own text onto this card; MtG and Parcly's universe of MLPFIM are both owned by Hasbro.

• That would be the most complicated card to ever exist XD Commented Dec 27, 2021 at 11:43

Yes, Tori must lose. Every $$5$$-coloring of $$\{1,\dots,161\}$$ contains a monochromatic $$(x,y,x+y)$$.
In fact, $$161$$ is the smallest such number for $$5$$ colors. See https://oeis.org/A030126.