A unusual feature of this contest is that it is to take place on “on a level, industrial type, carpeted floor”, contradicting the usual approach of minimizing rolling resistance by using very thin wheels. Another is that the car must carry a 1 lb (0.45 kg) load, contradicting the usual approach of making it very light. As with most contests of this sort, all cars have the same amount of available energy (possibly – see below), using the same type of fairly small rubber band.
Let’s look at the basic mechanics of this.
The amount of energy the car takes to move a distance

over the carpet is given by the formula

, where

is the force required to move keep the car moving, and is the opposite force to the force of friction. As a general rule, rolling friction is proportional to speed, so the faster the car goes, the shorter the distance

it will go.
So the design goal should be to have the car move as slowly as possible, without stalling (stopping) due to some small obstacle or imperfection in the carpeted floor or its wheels or bearings.
This force will come, of course, from the rubber band (or possibly some more complicated scheme where energy from the rubber band is stored in another form, such as a spinning flywheel, though since every transformation of energy in a real machine involved some loss due to friction, I’d be inclined to Keep It Simple). The rubber band, however, provides a varying amount of force at different stretched lengths, making things more complicated.
Good first steps in the design process are to measure the forces involved
For

, this means building the car without the rubber band drive, putting a 1 pound mass on it, and pulling or pushing it across a carpeted floor as identical to the one to be used in the competition as possible with some sort of force-measuring device – a scale, in other words, the more sensitive, the better. Since most scales are made to measure vertical, not horizontal force, getting a working test rig may involve a bit of a construction itself.
For the rubber band’s force

, you need to get a plot of it at different stretched lengths. You could use a scale for this, too, of reverse the process and hang weights of known mass from the band and measure how far it stretches.
Once you know the force of the rubber band and the force required by the car, you can calculate the necessary size of the transmission spool

and the wheels

. For the car to move,

. How much greater? Make a guess, then test it.
Here’s the fun and complicated part: remember that, as the rubber band pulls the string to turn the spool, axle, and wheels, if gets shorter, and the force it exerts decreases. So, to get the most efficient transmission,

needs to vary as the thread is pulled from it.
There are several ways to do this, some involving gears, some not. You could go with a fixed-ratio transmission, avoiding this complication, but if you do, you’re likely to lose to a more efficient design.
Another primary design goal is for the car to minimize

, by making the axle bearings as well as you can (ball bearings, etc), large wheels, etc. A truly weird configuration might do well – “thinking outside the box”, combined with making and testing lots or prototype models could allow you to hone-in on a winning design.
Last, keep in mind a rubber bands can store energy other than by being stretched lengthwise – they can be twisted. Figuring out the best way to use the band will take imagination and experimentation. It’s possible, if the rules permit it, that conditioning the rubber band with pre-stretching, oils or balms can increase the energy it can store. Other tricks are likely possible, and within the rules.
I imagine the winning car will be an interesting, and possibly surprisingly weird machine. Best of luck to you, hongher – may the winning car be yours (or your team’s)!

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