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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.

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    $\begingroup$ That would be the most complicated card to ever exist XD $\endgroup$
    – justhalf
    Commented Dec 27, 2021 at 11:43

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The answer is:

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.

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  • $\begingroup$ rot13(Naq urapr vg jnf pnyyrq Znevwa'f Oyhss – sbe gur bar jub cebirq vg.) $\endgroup$
    – Parcly Taxel
    Commented Dec 27, 2021 at 16:29

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