Why is 102,981,500 wrong?

This number was computed at LEGO in 1974
(snipped - from the web)

This number is - roughly - the number of ways one can build a tower of height six with two-by-four studded LEGO bricks. But when one builds with LEGO one does not have to put each brick on top of the previous one. Indeed, it is a main feature of LEGO that one can combine the bricks much more freely.

see http://www.math.ku.dk/index.html.en

Of current interest: LEGO counting problem

Why is this interesting?

Clearly nobody really needs to know exactly how many ways there are to combine 6 or more LEGOs, as clearly demonstrated by the fact that LEGO has done very nicely with the smaller number over several decades!

Thus this is mainly interesting as a challenge for mathematicians:
To compute or at least estimate the number of ways to combine a given number of two-by-four LEGO bricks.

Such challenges are always important to drive mathematical research, and it oftens happens that methods developed to study a problem with no practical applications (like this one) are useful to study problems which do have an impact on everyday life.

Mathematicians often get the question of whether there is anything left to study in mathematics.

The common misconception that the study of mathematics is somehow complete is probably induced by the fact that the mathematics most people encounter during their education is several centuries old.

The fact of the matter is that mathematics is a vivacious research area with lots of open problem, and here is a good one to point out.

thanks to Bergfinnur Durhuus, Søren Eilers and Mikkel Abrahamsen