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For which $n\in \mathbb{N}$ can we divide the set $\{1,2,3,\ldots,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$?

Since $x_i+y_i=3z_i$ for each subset $A_i=\{x_i,y_i,z_i\}$, we have $$4\sum _{i=1}^n z_i=\sum _{i=1}^{3n}i = {3n(3n+1)\over 2} \implies 8\mid n(3n+1) $$ so $n=8k$ or $n=8k-3$. Now it is not difficult to see that if $k=1$ we have such partition.

  • For $n=5$ we have: $$A_1= \{9,12,15\}, A_2= \{4,6,14\}, A_3= \{2,5,13\}, \\A_4= \{10,7,11\}, A_5= \{1,3,8\}$$
  • For $n=8$ we have: $$A_1= \{24,21,15\}, A_2= \{23,19,14\}, A_3= \{22,2,8\}, A_4= \{20,1,7\}, \\A_5= \{17,16,11\}, A_6= \{18,12,10\}, A_7= \{13,5,6\}, A_8= \{9,3,4\}$$

What about for $k\geq 2$? Some clever induction step? Or some ''well'' known configuration?

Source: Serbia 1983, municipal round, 3. grade

nonuser
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    I wrote a script using Knuth's dancing links to check this. For all $n\le 48$ for which $n\equiv 0\text{ or }5\pmod 8$, my script found a solution in a couple of seconds, **except** for $n=45$. After 10 minutes, I gave up. This doesn't prove there is no solution for 45, but it suggests it. (Here are the solutions I found: https://pastebin.com/MyYaPd6t). – Mike Earnest Jun 23 '21 at 18:28
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    I wrote a blog post that might be of interest: Richard Guy studied the analog where $x + y = 2z$. https://blog.peterkagey.com/2021/05/richard-guys-partition-sequence/ – Peter Kagey Jun 23 '21 at 19:21
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    I found one for $45$ with simmulated annealing. – Asinomás Jun 23 '21 at 20:05
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    what a nice question aqua! Might be easier to think of the condition as $\frac{\text{sum}(A_i)}{4} \in A_i$. – mathworker21 Jun 24 '21 at 23:02

2 Answers2

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If there is a solution for $N$, then there is a solution for $7N+5$.
The solution for $N$ uses up numbers from $1$ to $3N$. Then $$(3N+k, 15N+9+2k, 6N+3+k), k=1..3N+3\\ (12N+8+k,15N+10+2k,9N+6+k), k=1..3N+2$$ sits the numbers from $3N+1$ to $21N+15$ on top of them.

A similar method gives a solution for $25N+8Q$, for all $-13\le Q\le11$, whenever there is a solution for $N\ge 13$. Together with @RobPratt's solution, that covers all $N=8M$ and all $N=8M-3$.

I have started a new question for a different version at Split $\{1,2,...,3n\}$ into triples with $x+y=4z$ and also Split $\{1,...,3n\}$ into triples with $x+y=5z$ - no solutions?

Empy2
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  • does this solve the problem, with the computer evidence we already have? – mathworker21 Jun 30 '21 at 04:33
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    No, but it means there are arbitrarily large solutions. – Empy2 Jun 30 '21 at 04:50
  • Can you please post you update answer as new question with linked here. Perhaps someone will have new idea with this generalzation of yours. – nonuser Jul 04 '21 at 15:29
  • Aren’t the numbers $15N+12,…21N+14$ used up twice, one for each of the two types of subsets? – Tavish Jul 04 '21 at 18:08
  • The +2k means that One type has odd numbers,, the other has even ones. – Empy2 Jul 04 '21 at 19:17
  • @Empy2 Oops, my bad. – Tavish Jul 04 '21 at 19:31
  • @Empy2 the two different versions you asked are not the correct ones. For example, it's supposed to be subsets of $[5n]$ with $v+w+x+y = 5z$, rather than subsets of $[3n]$. – mathworker21 Jul 11 '21 at 20:24
  • For x+y=1z, build a solution for 4n+1 from a solution for n: Double the numbers in the solution for n. That uses even numbers up to 6n. Then x are odd numbers from 1 to 6n+1, y are descending from 9n+2 to 6n+2, z are increasing from 9n+3 to 12n+3. – Empy2 Jul 15 '21 at 11:17
  • For $x+y=3z$, when $3N=5m+2$, you can put the numbers from $5m+3$ to $11m+5$ in the same way. – Empy2 Oct 04 '21 at 13:07
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Here is the integer linear programming approach I used to find partitions for all such $n\le 496$ with $n \equiv 0,5 \pmod 8$. First enumerate all triples $\{x,y,z\}$ with $x+y=3z$ and $x,y,z$ distinct elements of $[3n]:=\{1,\dots,3n\}$. For each such triple $T$, let binary decision variable $u_T$ indicate whether $T$ appears in the partition. The constraints $$\sum_{T:\ i\in T} u_T = 1 \quad \text{for $i\in[3n]$} \tag1$$ enforce that each element appears exactly once in the partition.

An alternative approach is to introduce nonnegative slack variables $s_i$, replace the set partitioning constraints $(1)$ with (set covering and cardinality) constraints \begin{align} \sum_{T:\ i\in T} u_T + s_i &\ge 1 &&\text{for $i\in[3n]$} \tag2 \\ \sum_T u_T &= n \tag3 \end{align} and minimize $\sum_{i=1}^{3n} s_i$. A partition of $[3n]$ into $n$ triples with $x+y=3z$ exists if and only if the optimal objective value is $0$.

RobPratt
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  • Thanks! Can you check about partitioning $\{1,2,\dots,5n\}$ into $n$ subsets each of size $5$ so that in each subset $\{v,w,x,y,z\}$ we have $v+w+x+y = 5z$? Of course, only certain $n$ work. – mathworker21 Jun 29 '21 at 23:43
  • @mathworker21 Your $5n$ variant requires that $12 \mid 5n(5n+1)$, so $n \equiv 0,3,4,7 \pmod{12}$. There are solutions for all eight such values of $n$ up through $n=24$. – RobPratt Jun 30 '21 at 03:17
  • Thank you very much. That's very helpful to know. – mathworker21 Jun 30 '21 at 03:18