“Tormenting Gemini 2.5 with the [[[]]][][[]] Puzzle” by Czynski
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Includes a link to the full Gemini conversation, but setting the stage first:

There is a puzzle. If you are still on Tumblr, or live in Berkeley where I can and have inflicted it on you in person[1], you may have seen it. There's a series of arcane runes, composed entirely of []- in some combination, and the runes - the strings of characters - are all assigned to numbers; mostly integers, but some fractions and integer roots. You are promised that this is a notation, and it's a genuine promise.

1 = []
2 = [[]]
3 = [][[]]
4 = [[[]]]
12 = [[[]]][[]]
0 =
-1 = -[]
19 = [][][][][][][][[]]
20 = [[[]]][][[]]
-2 = -[[]]
1/2 = [-[]]
sqrt(2) = [[-[]]]
72^1/6 = [[-[]]][[][-[]]]
5/4 = [-[[]]][][[]]
84 = [[[]]][[]][][[]]
25/24 = [-[][[]]][-[]][[[]]]

If you enjoy math puzzles, try it out. This post will [...]

First published: March 29th, 2025

Source: https://www.lesswrong.com/posts/xrvEzdMLfjpizk8u4/tormenting-gemini-2-5-with-the-puzzle)

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Narrated by TYPE III AUDIO).