IRL_ABooklet_2023

For each $N>1$, let $A_N$ denote the set of all positive integers that are not multiples of $N$, and let $a_{N,1}<a_{N,2}<a_{N,3}<\dots$ be the elements of $A_N$ in increasing order. The set $A_N$ consists of blocks of $N-1$ consecutive integers followed by a skipped integer. If $N\equiv 1(\mathrm{mod}3)$ the length of each of these blocks is a multiple of 3 and so $a_{N,i}$ can only be the last element of a block if $i$ is divisible by 3. In other words: $$\text{If }N\equiv 1(\mathrm{mod}3)\text{ and }i\ne 0(\mathrm{mod}3),\text{ then }{a}_{N,i+1}=a_{N,i}+1.(1)$$ For each $N>1$ that satisfies $N\equiv 1(\mathrm{mod}3)$, we will show that $A_N$ is a special set. Since $2023\equiv 1(\mathrm{mod}3)$, the desired result follows. We now fix $N>1$ satisfying $N\equiv 1(\mathrm{mod}3)$. For simplicity, from now on we suppress the $N$ index in $a_{N,i}$. We will use induction to show that for each $n\ge 3$, the numbers $a_1,a_2,a_3,\dots ,a_{3n-3}$ can be used to form a special $n$-triangle. The base case consists in showing that there are special 3- and 4-triangles. $$\begin{array}{ccccccc}& a_1& & a_1& & & \\ & a_6& a_5& & a_6& a_9& \\ a_2& a_4& a_3& a_7& & a_5& \\ & & & a_3& a_4& a_8& a_2\end{array}$$

When $N\ge 10$, $a_i=i$ for $1\le i\le 9$, and it is easily checked that these triangles are special. For $N=4$ the above turns into and when $N=7$ we get and these are indeed special triangles. For the inductive step, we show how to obtain a special $n$-triangle for $A_N$ from a special $(n-2)$-triangle for $A_N$. Any special $(n-2)$-triangle contains the numbers $a_1,a_2,\dots ,a_{3n-9}$. We group the next six numbers of $A_N$ into two sets, $S_1=\{a_{3n-8},a_{3n-7},a_{3n-6}\}$ and $S_2=\{a_{3n-5},a_{3n-4},a_{3n-3}\}$. By virtue of statement (1), both sets consist of three consecutive integers. Hence, $$a_{3n-3}+a_{3n-8}=a_{3n-4}+a_{3n-7}=a_{3n-5}+a_{3n-6}.$$ It is now clear that we obtain a special $n$-triangle when we add $a_{3n-3}$ and $a_{3n-8}$ to one side, $a_{3n-4}$ and $a_{3n-7}$ to another, and $a_{3n-5}$ and $a_{3n-6}$ to the final side in any given special $(n-2)$-triangle.