The missing levels require ...
... 41 003 010 and 40 000 000 bricks.
With these numbers, the total sum of all bricks in the tower is 281 828 172, which are the first nine decimal digits of Euler's number e = 2.718281828459..., read from right to left.
The sum of the digits on each level are the digits of π = 3.141592653..., read from bottom to top. (I hadn't realised this until JLee pointed it out.)
{Op edit: This is equivalent to right-aligning the numbers and then summing the columns from right to left to construct a number, and summing the rows from bottom to top to construct a number. This insight is important for the later bonus challenge). Also note that finding one # without the other does not lead to a unique solution.}
Here's the complete tower with the zeros left out for clarity:
· · · · · · · 3 · 3
2 · · · · 3 · · · 5
· · · 4 2 · · · · 6
· · · · · 1 1 · · 2
· 4 1 · · 3 · 1 · 9
· · · 3 · · · · 2 5
· · · · · 1 · · · 1
· 4 · · · · · · · 4
· · · 1 · · · · · 1
· · · · · · · 3 · 3
2 8 1 8 2 8 1 7 2
That was hinted at ...
... by the fact that a mathematician had commissioned the tower. Leonhard Euler was a famous mathematician. And Euler's formula, which uses both e and π is considered to be very beautiful by some, also in our circles.
The grey girders, when flipped horizontally, look like an e, so the digits of e must be read right to left.. When flipped vertically, they look like a π, so π must be read from bottom to top.
Bonus challenge: A golden tower ...
... is related to the first nine digits of the golden ratio φ = 1.61803398874..., so we must arrange the existing digits horizontally to get 161 803 398, whose digit sum is also 39. But I don't see how.
Op Edit: (self-completing as the difficulty level of the bonus challenge was beyond the scope of the main puzzle.)
We have seen that if we align the digits to the right:
The sums of the columns from right to left form the beautiful number e.
The sums of the rows from bottom to top form the beautiful number pi.
So how can we form a third beautiful number without changing any integer values or the floor they are on?
Well, we were asked to make a golden tower, not a nedlog one, so let's align the digits to the left and sum left to right!
But we face a problem! The left digits sum to 26, and we are looking for the first 9 digits of phi, so it should begin with 1 or 16.
The secret here is to notice that we can left-pad zeros to the numbers; this will change their left alignment but not their integer values!
(Or you could just stagger them with whitespace, for the same result)...
000000030 | 3
0200003000 | 5
0000000420000 | 6
00000001100 | 2
041003010 | 9
000300002 | 5
0001000 | 1
00040000000 | 4
100000 | 1
0000000030 | 3
_____________
1618033980000
Note that due to similar rows with a sole 1 or 3, a few trivial variants are possible by swapping similar rows, but they are essentially equivalent constructions)
The rows still add up to pi, the right-aligned digits still add up to e, and now the left-aligned rows add up to
1618033980000, which if we insert the required decimal, is 1.618033980000, which is mathematically equal to 1.61803398, the first nine digits of the golden ratio, as desired.
Thus our golden tower, concealing all three beautiful numbers, is complete!
