# ETOOBUSY ðŸš€ minimal blogging for the impatient

# Autobiographical numbers constraints - step up

**TL;DR**

Where we remove all programming joy from this nice puzzle.

Do you see any pattern?

```
$ for N in $(seq 10 20) ; do ./run.sh 04-luckier-sum "$N" ; done
solution => [6,2,1,0,0,0,1,0,0,0]
solution => [7,2,1,0,0,0,0,1,0,0,0]
solution => [8,2,1,0,0,0,0,0,1,0,0,0]
solution => [9,2,1,0,0,0,0,0,0,1,0,0,0]
solution => [10,2,1,0,0,0,0,0,0,0,1,0,0,0]
solution => [11,2,1,0,0,0,0,0,0,0,0,1,0,0,0]
solution => [12,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0]
solution => [13,2,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
solution => [14,2,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
solution => [15,2,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
solution => [16,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
```

It *seems* that this would always be a solution, at least for $N$
*sufficiently large*:

`0`

contains value $N - 4$`1`

contains value $2$`2`

contains value $1$`N-4`

contains value $1$- everything else is $0$.

When $N > 6$, then $N - 4 > 2$ which is the condition in which slot `N-4`

does not overlap with any of the other three slots that have non-zero
values.

Is this *always* a solution for $N > 6$ a.k.a. $N - 4 > 2$? Yes it is:

`0`

,`1`

,`2`

, and`N-1`

are 4 distinct slots, because $N-4>2$;- these are the only slots holding a value different from $0$;
- all the other slots (i.e. $N - 4$ of all slots) hold value $0$, which is
consistent with the value at slot
`0`

; - value $1$ appears exactly 2 times (in slot
`2`

and`N-4`

), and slot`1`

contains value $2$; - value $2$ appears exactly once (in slot
`1`

), and slot`2`

contains value $1$; - value $N-4$ appears exactly once (in slot
`0`

), and slot`N-4`

contains value $1$.

So thereâ€™s no need for complicated searches for $N > 6$: just provide the solution according to the pattern above.

```
sub autobiographical_numbers ($n) {
my @solution;
if ($n == 4) {
@solution = (1, 2, 1, 0); # also good: (2, 0, 2, 0)
}
elsif ($n > 6) {
@solution = (0) x $n;
@solution[0, 1, 2, $n - 4] = ($n - 4, 2, 1, 1);
}
return {solution => [map {+{$_ => 1}} @solution]};
}
```

Find all of this at stage 5.

How boring. And yetâ€¦ are these the *only* solutions?!? E.g. $N = 4$ allows
two different solutionsâ€¦ is it possible elsewhere?!?

# The end of it

Curious about the whole series? Here it is:

- Autobiographical numbers
- Autobiographical numbers constraints - basic
- Autobiographical numbers constraints - last is zero
- Autobiographical numbers constraints - weighted sum
- Autobiographical numbers constraints - luckier weighted sum
- Autobiographical numbers - step up
- Code repository

Comments? Please comment below!