In the free group generated by four letters, if you remove any two letters (and their ...

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npub12c…gd38p

2026-09-06 05:36:19 UTC

In the free group generated by four letters, if you remove any two letters (and their inverse) from this product of 16 elements:

𝑎𝑏𝑎⁻¹𝑏⁻¹𝑐𝑑𝑏𝑎𝑑⁻¹𝑐⁻¹𝑑𝑐𝑏⁻¹𝑎⁻¹𝑐⁻¹𝑑⁻¹

it simplifies to 1, but dropping any single letter does not give 1.

There are no shorter products with this property.

This is an example of the famous “picture hanging puzzle”: if you have N nails in the wall, what are the least complicated ways to wind a string holding up a picture around the nails so that removing any K of the nails will cause the picture to fall, but removing fewer than K nails will leave the picture supported.

The result above is from recent work by Tom Verhoeff:

https://arxiv.org/abs/2605.27617

that I read about in Scientific American:

https://www.scientificamerican.com/article/mathematicians-discover-the-worst-way-to-hang-a-painting/

From the paper: Demaine et al.’s general solution gave a word of length O(N^{1561600}) [yeow!] Wästlund improved this to O(N^{7.004}).

Author Public Key

npub12cuzzqzv8e8y7na77a9zv47mx2v6pec60qs5h7takzmv5p0mw0fsmgd38p

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wss://relay.momostr.pink

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Published at

2026-09-06 05:36:19 UTC

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1 Short Text Note

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