Can anyone tell me how they came to this conclusion? I’m somewhat familiar with combinatorics, but find this figure hard to believe.
FWIW, I believe that figure refers specifically to six 2x4 bricks.
(Incidentally, that page is the top Google result for “915103765” 
You’ve got me stumped. For a combinatoric number, 915103765 is remarkably short of factors: The only factor it has under 110 (as far as I was willing to check by hand) is 5. If nothing else, this means that they’re treating the bricks as indistinguishable, because otherwise, the number would have to be a multiple of 720. And there are enough ways to fit them together (and enough qualitatively different ways, once you get above 2) that I have a hard time seeing how to calculate this, other than a brute-force count. Even that would be difficult, though, because there would be a lot of degeneracies.
From a brief skim through the page I linked to, I guess that is because it is a sum of separate combinatorial numbers – for towers of heights 2, 3, 4, 5 and 6. See about halfway down the page:
**Height Number**
2 7,946,227
3 162,216,127
4 359,949,655
5 282,010,252
6 102,981,504
Total 915,103,765
Strictly speaking, they’re not even calculating the full number of arrangements, as you don’t have to connect all the blocks. If L(k) denotes the number of ways to arrange k blocks in a single tower, the total number here should be L(6) + L(5)L(1) + L(4)L(2) + L(4)L(1)[sup]2[/sup] + L(3)[sup]2[/sup] + L(3)L(2)L(1) + L(3)L(1)[sup]3[/sup] + L(2)[sup]3[/sup] + L(2)[sup]2[/sup]L(1)[sup]2[/sup] + L(2)L(1)[sup]4[/sup] + L(1)[sup]6[/sup]. Since L(1) = 1, this reduces to L(6) + L(5) + L(4)L(2) + L(4) + L(3)[sup]2[/sup] + L(3)L(2) + L(3) + L(2)[sup]3[/sup] + L(2)[sup]2[/sup] + L(2) + 1. We also know that L(6) = 915103765, so we have only a few more values to calculate.
It wasn’t until deep in a page linked from one of the above references that I found the specification of A TOWER, or one brick on top of another. Here I was picturing “arrange six bricks” as including formations such as a flat L (interlocked in two layers), a hexagon (each brick connected at one corner), and so on.
Bet that adds to the total, unless we are sticking with the restriction of a tower only.
Doesn’t that page say that the earlier, lower, claimed number (from LEGO itself) was for a tower 6 high, whereas the newer, higher, number takes into account arrangements 2 layers high, 3 layers high and so on?
By the way, does the expression “six Legos” grate on anyone else as much as it does me? Who the heck calls them “Legos”? Pieces of Lego, bits of Lego, Lego bricks, yes, but not Legos!
Don’t let some corporate marketing drone tell you how to speak. Call them legos like normal people do.
Lots of people in the US, but they’re wrong, according to the Lego company. Yes, I know that they don’t actually have any authority over language usage, but after all the many hours of enjoyment I’ve gotten from their products, I figure using the terminolgy they stipulate is the least I can do in return.
It has nothing to do with “corporate marketing drones”. When I was a wee nipper I had a big box of Lego. Not Legos. Same as out in the garden I had a pit full of sand, not sands. Lego just seems much more comfortable as a mass noun… like Meccano.
But, if it’s a UK/US divide then who am I to argue. (Although I will say, what the heck is “savings” as a singular noun all about? 
)
Yes, but at the very top of the page it only mentions combining the bricks, and you have to kind of wade through the explanations to discover that only a tower (of any height) is the subject of the problem.
Oh, and:
If you count this, then there are an infinite number of combinations, with only two bricks. Two bricks connected only at a single stud can be at any angle within some range, and there are an infinite number of angles. A finite answer must necessarily presume some limited selection of angles, and the most logical set of angles to use is 0, 90, 180, and 270 degrees.
Yes, this is a US usage versus European usage thing. In the US, “Legos” is standard. It’s rare to hear them called anything else.
OK, fair enough. But I still think that just “combining” at variations on 90 degrees would also allow for a multiple-level structure, with the bricks going up and down in relation to each other like steps, rather than just a simple vertical succession.
I’m probably muddying the waters here, just trying to show that I think the problem statement could be clearer.
No, not necessarily. We can look at which connections are made, disregarding the angles they make. With two bricks connected by a single stud, over the range of possible angles, you’ve only got the one connection, so they’re all equivalent to the multiple-of-90-degrees case. A hexagon shape is connected in a way that is impossible using only angles that are multiples of 90 degrees*.
- in Euclidean space, and with unmelted, unbroken bricks, before some joker pipes up.
You were lucky! I had to make do with an empty box and no sand in the pit.
It has everything to do with corporate marketing drones they just got to you at an early age. Legos as a mass noun comes from the lego company.
I don’t know how you interact with the sand in the sand box but I always interact with in a large number of grains at a time. In contrast to legos. With legos you generally have a need to refer to individual lego blocks. Most of the mass nouns that I can think of you don’t really interact with them at the individual unit level.
I’d have called 'em chazwozzers!
It’s “lego” here too, FWIW.