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In this question, bridge stands for the illustrated convex quadrilateral, constructed from three equilateral triangles. Letters A, B, C, D, E are assigned to each unit length of periphery as shown in the diagram. The dot indicates the location of the center of gravity (COG) of a bridge; when the origin is at lower left, the COG is at (x,y) = $(1, 2\sqrt 3/9)$. An assembly of bridges has a lowermost horizontal base line that is a face of its first bridge.

protobridge picture

Rules: We allow a labelled side on a bridge to connect exactly to a labelled side on another bridge. Bridges cannot overlap. As pieces are added to an assembly, it must never tip over; ie the x-coordinate of the progressive COG must always fall within or on the base line.

3 example structures

In the diagrams, numbers from 0 to 4 or 5 denote the order in which bridges were added. The path of small asterisks in each figure represents the path of the COG as parts are added. The large asterisk shows the final COG. For example, in the right-hand diagram, the COG first moves to the left as parts 1 and 2 are added, and back to the right as parts 3 and 4 are added.

Objectives: Among valid assemblies of 7, 8, or 9 bridges, find those that have the highest ratio of overall width to baseline length. (Left to right, the figures shown have ratios 4.5/2 = 2.25, 5/1 5, and 4/2 = 2.)

Notes: (1) The COG of an assembly is the area-weighted average of the COGs of its parts. (Ref: wikipedia) (2) The diagrams were generated as PS files by a slightly clunky python program that I've uploaded to pastebin.

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  • $\begingroup$ I'm not quite sure I understand the criterion of 'tipping over'. If I understand correctly, you mean the overall COG should never be outside of the created structure? (as in: the construction needs to be able to rest on a surface directly below the COG and therefor not tip over?). Because that would imply the entire structure shifting relative to the 'holding point', rather then real life bridges where the base point always remains the same, and the COG would always need to be within 'bridge 0'. $\endgroup$
    – Tim Couwelier
    Aug 22, 2014 at 8:25
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    $\begingroup$ Oh wait, the term 'bridge' had me confused. I was looking at the problem as the top view of an actual bridge section, whereas you built it as a 'tower'. As long as the COG is situated above edges D or E of 'bridge 0' it will not tip over. $\endgroup$
    – Tim Couwelier
    Aug 22, 2014 at 8:27
  • $\begingroup$ Here is a grid for "MS-Painting" solutions: i.imgur.com/tIdvvkP.png $\endgroup$
    – aaaaaaaaaaaa
    Aug 22, 2014 at 9:32

1 Answer 1

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There doesn't seem to be much challenge in this, from a simple base the construction can be extended indefinitely by alternating between adding parts to the right and left side:

Easy solution

This gives solutions of width 10, 11.5 and 13 for 7, 8 and 9 pieces respectively.

I by the way find it easier to calculate the centre of mass by considering each triangle separately, that way all the involved numbers are divisible by 1/2.

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  • $\begingroup$ If that works, why not this instead? i.imgur.com/18ksies.png $\endgroup$
    – Tim Couwelier
    Aug 22, 2014 at 11:20
  • $\begingroup$ @TimCouwelier: That has a large base width. $\endgroup$
    – Ross Millikan
    Aug 22, 2014 at 14:06
  • $\begingroup$ @TimCouwelier, I now see that baseline length or base width isn't well-defined in the problem. But the idea is that eB's figures have ratios 10, 11.5, and 13, while your flat figures should have ratios of about 1. BTW, I edited my program to have a % comment with COG location $\endgroup$
    – James Waldby - jwpat7
    Aug 22, 2014 at 14:37
  • $\begingroup$ @eBusiness, How did you draw this? $\endgroup$
    – Rafe
    Oct 8, 2014 at 17:27
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    $\begingroup$ @Rafe Using MS Paint, draw a few lines, copy them, insert, and place them in a grid formation, repeat until a suitable area is covered. Then flood fill and insert numbers with the text tool. $\endgroup$
    – aaaaaaaaaaaa
    Oct 8, 2014 at 17:40

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