Figure 4: Upper bound on bricks per layer, conical Lego cup, computed three different ways. The blue line is the theoretical maximum, using equation (1) and is a strict
upper bound. The red line considers the size of the largest square that could be fit at
each layer and should be considered a strict lower bound. The green line is the linear
trend line of the brick counts of the “bottom” and “top” layers, pictured in Figure 3.
This equation may be readily implemented in Excel. Because there is
a “floor” function on the summands,
for our purposes need only be evaluated up to
. The base of the cup
has a radius of approximately 4.4 LC
and has a theoretical maximum of 61
bricks. I was able to achieve a base
layer of 58, but this is probably because I’m not a great builder. The top
of the cup has a radius of approximately 6 LC and has a theoretical maximum
of 113 Bricks. I was able to achieve 98
A NA L Y T I C S
in my build. Using these and assuming
a linear trend in the cup (the sides of
the cup look smooth and straight), we
estimate that a theoretical maximum
of 1,364 Lego bricks could fit in the
round cup, with a more likely number
being approximately 1,250. See Figure 4 for three different calculations of
bricks-per-layer.
So the round cup holds about 200
more bricks than the box if you take the
time to pack it. Real Lego enthusiasts
use a greedy heuristic to fill their cups,
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