Let $ [n] $ be the set $ \{1,2,\ldots n\}$.

A *summoid* is a subset $ A \subset [n] $ of the form $ \{a,b,a+b\} $ (you can choose a better name, if it doesn't exist already).

Now, I developed by accident this simple result:

There is no partition of $ [9] $ into (disjoint) summoids.

I want to ask the following questions:

Is it true for general $ [3k] $ when $ k > 1 $? According to a computer program I found that it is true for $ [12] $, but it seems my method for $ [9]$ can't be applied to the general case (maybe my program for $[12]$ isn't trustworthy).

**According to the comment below by R. van Dobben de Bruyn**, there is no such partition when $ k \equiv 2,3 \pmod 4$.**According to the comment below by Gerhard Paseman**, there is a counterexample for $ k = 5 $ that extends also to $ n = 3 \cdot 4^k $ and to $ n = 3 \cdot (1+4^k) $.Is it useful for something?

- Is there any study of such results?