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 there’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 yet… are these the only solutions?!? E.g. $N = 4$ allows two different solutions… is it possible elsewhere?!?

# The end of it

Curious about the whole series? Here it is: