This is a house-of-cards style challenge in which the objective is not
to build the highest tower but the one with the largest unsupported
span. In principle it's possible to simply stack playing cards face up/
down to produce an arbitrarily large overhang. i.e. the top card can
be laterally displaced
from the bottom card by some distance less
than or equal to infinity [insert lengthscale here]. Although the
profile of an optimised stack can be derived mathematically,
micron-scale manual precision will be required when placing the
cards. Pre-fabricated construction aids such as formers and
scaffolding are allowed to help erect the tower but these supporting
structures must be removed prior to measuring the final
unsupported span. A realistic overhang of three card lengths should
be do-able with around 2000 playing cards, giving a tower
approximately 500mm tall. Not suitable for outdoors.

I don't think the structure you refer to is very
interesting - the length of cantilever is limited
only by the number of cards you have and how
precisely you can place them.

But the idea of a competition for maximum
unsupported span is a good one, assuming that
entrants have free choice as to the design. It
should also be limited to the standard deck of 54
cards.

In theory (and only in theory) the widest span
would be almost 54 cards wide - an shallow arch
consisting of cards placed end-to-end (like a
normal stone arch, but with the stones being
incredibly thin top to bottom). However, this
structure would be at best meta-stable.

You might be able to do something similar, but
only a little less than 18 cards wide, by building an
arch in which each segment ("stone") consisted of
three playing cards in the form of a triangular
prism (axis running along the arch). Alternate
triangular segments would be rotated 180°
relative to eachother, providing a reasonably
stable structure. I think this could actually be
done.

“Bridge” suggests a span touching the table at each end; my initial impression was that we were looking for an “overhang” supported at one end only.

[MB] a shallow arch needs to push outwards at the ends - what do the rules say about the nature of the table surface or the use of weighted blocks as buttresses? I am guessing they are not allowed.

Also your prism design would not be stable because the inner cards wouldn't be shorter than the outer ones.

A much stronger 'building block' could be made by making a short tear halfway down the left and right side of a normal playing card, rolling the card into a tube and slotting these tears into each other. Then, these cylinders can be joined end-to-end to form a 'pipe', slightly less than 54 playing cards long.

This is crucial. If buttresses are banned then that implies that the bridge carries no axial load or bending moment at mid-span. If this is the case then the bridge is effectively two non-interacting cantilevered "overhangs" (as was the intent of the original idea) joined back-to-back which seems somewhat redundant.

I can see that a cantilever has limited design potential and will only ever acheive moderate spans (with finite resources), but buttresses feel like cheating. I'm undecided which I would rather see, but I suspect "bridge of cards" is baked.

OK, no buttresses it is. However, you would also
need to specify the surface on which the thing is
constructed - friction against a rough surface can
provide a lot of buttrefication.

So, a defined surface, and no folding, cutting or
sticking. You'd have an issue over the definition of
"folding". For instance, putting a slight bow in a card
gives it vastly more regidity at right angles to the
bow.

What's the formula for span vs. number of cards
(perfectly) stacked? If the n-th card added to the
base of the stack adds 1/(2*n) card lengths to the
span, then you can span an infinite distance with
an infinite stack of cards, but the stack will be
much taller than it is wide.

As you say [sninctown] the offset between successive cards is 1/(kn), where n is the number of cards and k is a kind of stability factor which must be >=2 for positive stability. I concur with your calculations. Part of the challenge would be deciding on the optimum value of k, thereby trading stability for height.

//surface// Something dead flat and stable. A 4" thick steel precision measurement table (the kind used for CMM measurements) springs to mind. Every home should have one. To deter people from roughening the surface to gain a frictional advantage the first card could be stuck to the table and then all cards have to be placed above that.

//"folding"// One solution to this might be that all cards have to succesfully pass through an automated card shuffler after dismantling the build. You could think of it as a post-race drugs test for playing cards.

I think that if contestants stick rigidly to this set of rules we can pretty much guarantee that no fun will be had by all.

I think a smooth working surface would be too detrimental to progress. Something more like a pool table would be better - flat and rigid enough, but the fabric covering would allow card edges to "grip", facilitating construction (or perhaps more sand-paper-like?). But not external butresses or other supports, just cards. I like it!

Not having a pack of cards in the house, I experimented with 10 CD cases. I found that a genuinely cantilevered structure, ie. 1 case on the first level, 2 on the 2nd, with a counterweight on top etc. it was much easier to build a quick overhang than using a single exponential overhang. My guess is that the single stack has a final result higher, but the more symmetrical cantilever is more tolerant. So I think if there were also a time limit (perhaps 1 minute) the game might become more fun as well as encouraging alternative designs, trading theoretical maximum against ease and speed of construction.

Who would down-vote this? Fess up y' tarot-phobic fish-carcass slinger blighter!

There should be unlimited classes but based on exact multiples of fifty two. Adding weights to the center of the span after construction could also be a variable. Not the longest span of cards, but the strongest span would win. The same arch might even win in both categories, but it would all depend on the keystone.