Motivation for solution to constructing a set of 1983 distinct integers such that no three are consecutive terms of an arithmetic progression

Suppose you have come to the point where you know that you are looking for a set of numbers which does not contain a triplet $x$, $y$, $z$ such that $x+z=2y$.

If I give you three numbers, how do you check if they do or do not satisfy that equality? Well, the stupidest way is to simply see if $x$ and $z$ add up to $2y$, of course!

Now, if we pick two numbers and actually try to add them, we immediately notice that there is something screwing with us: all the carrying over from one column to the other. So, to avoid that, we only consider numbers $x$ and $z$ such that when we add them there is no carrying, crossing our fingers so that this condition is not too draconian to leave us with too few candidates... (And hey, we got the opportunity to use the word draconian!)

We also need to compute $2y$, so we may just as well —keeping our fingers crossed— assume that when we compute it there is also no carrying over.

If the digits of $x$ are $x_nx_{n-1}\cdots x_0$, and similarly for $y$ and $z$, at this point, the condition that $x+z=2y$ translates into

$x_i+z_i=2y_i$ for all $i\in\{0,\dots,n\}$.

So, what we want is that if the digits of $x$, $y$ and $z$ satisfy this condition, then in fact $x$, $y$ and $z$ cannot be all different. So, if we only allow the digits of out numbers to come from a set $S\subseteq\{0,1,\dots,9\}$, we want that

for all $a$, $b\in S$, then $a+b<10$

so that there is no carrying over when we compute $x+z$, nor when we compute $2y$, and that

if $a$, $b$, $c\in S$ are such that $a+b=2c$, then in fact $a=b=c$.

(This does look like more than what we really need, but I can't think of what we really need... if it does not work, this is were we need to think more...)

So... a little work will show that we can pick $S=\{0, 1, 3, 4\}$. and then it works. Cool!

So we only have to consider all numbers whose digits are drawn from ${0,1,3,4}$ and which are less than $100,000$. Hmm. We think a bit and see that there are too few of these! Damn. In fact, there are 1025 of them.

Start over.

Well... now we have a little idea: what is this obsession with the number $10$. Really. I never wrapped my head with using $A$ and $B$ and so on as digits, so well, I'll try to use another base, but smaller than $10$.

(Hmm, base $2$, the usual suspect, is not going to help here...)

Random pick: let's do base $6$. Our set $S$ of digits will have to be drawn from ${0,1,2}$, because for $3$ already we have carry over when doubling. Hm. The sets $S$ we can construct have at most $2$ elements; for example, $\{0,2\}$ or $\{1,2\}$. Hmm. Thinking a bit shows there are too few numbers smaller than $100000$ using those $6$adic numbers (we of course prefer $\{0,2\}$ to $\{1,2\}$, because it allows us to write smaller numbers, so more numbers). Damn again.

So... Think a bit more... Using base $5$ is not going to help, because the maximal digit is also $2$... Base $4$... Ok. We can take $S=\{0,1\}$, and work, work, work, there are only 512 numbers below $100000$ using only them. Ok, but base $3$ allows us to use the same digits, and obviously there will be more numbers with only those digits. Hmm. Ooooo. $2048$ of them!

We did it :)

if this is the number line:

ooooooooooooooooooooooooooooooooo...

the first dot represents 1, the next 2, ...

notice there's an arithmetic progression here:

ooooooooooooooooooooooooooooooooo...
^^^

we can initially remove all arithmetic progressions of step 1 by removing every third

oo oo oo oo oo oo oo oo oo oo oo ...
^^^

notice that it doesn't contain any arithmetic progressions of step 2 either! but it does contain an arithmetic progression of size 3:

oo oo oo oo oo oo oo oo oo oo oo ...
^  ^  ^
 ^  ^  ^

so let's remove the third from each of those

oo oo    oo oo    oo oo    oo oo ...

notice that the arithmetic progressions of size 4, 5, 6, 7 and 8 are missing too! The next one we have is:

oo oo    oo oo    oo oo    oo oo ...
^        ^        ^

so we can remove those..

oo oo    oo oo             oo oo ...

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I think that was a natural way to approach the problem and after actually doing it for a little while the pattern and connection with base 3 numbers is clear. So the next step is to write it out in mathematical symbols and remove all the scaffolding, that's how you end up back at the proof you posted.