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How many ways can you stack six 2x4 lego bricks? In 1974, LEGO said it was 102,981,500. But! High school student Mikkel Abrahamsen and mathematician Søren Eilers revisited the problem and got 915,103,765. Here's the paper [pdf] with the details and some nice graphs and illustrations. And if that's not enough for you, see also On the entropy of LEGO [pdf] by Eilers and Bergfinnur Durhuus.
posted by cortex (6 comments total) 13 users marked this as a favorite

Seems it's the problem definition that was fuzzy (telephone game?), not so much the original number. From the introducftion to the entropy paper:

However, [102,981,500] only gives (with a small error, as we shall see) the number of ways to build a tower of LEGO blocks of height six. The total number of configurations is 915,103,765.

Given a small number of identical bricks, Lego towers are obviously a more natural Lego construct than arbitrary abominations, so I'd say both results are important.

(and yeah, for the lazy, the small error is that the original number wasn't reported with enough significant figures, the exact value is 102,981,504, so about 0.000004% off)
posted by effbot at 12:13 PM on February 17, 2017

Lego towers are obviously a more natural Lego construct than arbitrary abominations,

Watching small children play with Lego suggests this may not be wholly accurate.
posted by howfar at 12:43 PM on February 17, 2017 [2 favorites]

I look forward to the inevitable video showing all of them.
posted by Segundus at 2:29 PM on February 17, 2017

Since these are 2x4 blocks, they don't account for my favorite way to stack Lego, which is, "off-grain." It turns out a 1x2 brick's studs are just thin enough that, with a little force, the studs can fit in the circular "holes" beneath a 2x4 brick. You can also do it with a 1x1 brick fitting into the single hole beneath a 2x2. These strange and unnatural Acts of Lego are a way to add a little variety to its strict, orderly universe.
posted by JHarris at 2:36 PM on February 17, 2017

Submitting something to the OEIS is a totally unreachable but constant dream for me.
posted by creade at 2:36 PM on February 17, 2017 [1 favorite]

It turns out a 1x2 brick's studs are just thin enough that, with a little force, the studs can fit in the circular "holes" beneath a 2x4 brick

You monster
posted by Salvor Hardin at 4:18 PM on February 17, 2017 [8 favorites]

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