# Building a pyramid from di-spheres

Suppose you want to build a square pyramid made up of 30 spheres. So the bottom layer is a 4x4 square arrangement of touching spheres, and it has 3x3, 2x2 and 1x1 layers on top.

The building blocks you have are 15 di-spheres, i.e. they consist of two spheres stuck together.

At all times while building the pyramid, you want every sphere to be fully supported underneath, either by four spheres in the layer below or by the ground.

Is it ever possible to get stuck, where there is no possible way to complete the pyramid without first removing pieces?

This question arose when someone I know was trying to design a two-player game. It is not a very difficult puzzle, but fun to work out.

• Is it okay to place a disphere between levels when the fourth support is part of the disphere you are currently placing? – boboquack Mar 27 '17 at 8:58
• @boboquack: Yes, of course. Otherwise you would not be able to place the tip of the pyramid. All that matters is that after placing any piece, all spheres must be fully supported. – Jaap Scherphuis Mar 27 '17 at 9:40

Does this arrangement work?

Fist layer:

AGBB
AEEH
IFFD
CCJD

Second layer:

.G.
I.H
.J.

Same letter denotes same disphere. The dispheres are placed in alphabetical order.

• Very good, especially for including the move order to get to that position. Bonus question: Is this the only possibility? – Jaap Scherphuis Mar 27 '17 at 11:01
• Well, the lowermost incomplete layer must always have a checkerboard pattern, otherwise it would either have two adjacent holes or three balls and one hole in a 2x2 square, both of which can be built upon. For parity reasons, this is the only good pattern we can build from dispheres, but I'm not sure if this is the only way to build it (ignoring symmetry). – BaSzAt Mar 27 '17 at 11:47

Maybe I solved something else, because of low understanding, but: 1st layer ('oo' 'o ↵ o' are stick together, 'O' goes up:

oo O o
O oo o
o oo O
o O oo

so third layer looks like ('x' empty):

x O x
O x O
x O x

and now if "you want every sphere to be fully supported underneath, either by four spheres in the layer below or by the ground" - there is no move now.
Because you can't make for example:

o O x
O o O
x O x

Or to go to another layer, because of "need support" rule.

• Unfortunately you were pipped to the post by BaSzAt, who posted his answer less than a minute before you did. So although your answer is perfectly fine, I've selected his as the accepted answer, and merely upvoted yours. Sorry! – Jaap Scherphuis Mar 27 '17 at 12:02