Very little is known about the physical properties of Kuiper-belt objects, due to their relatively small size and large distance from the Earth. For example, a Kuiper-belt object with a diameter of 300km at a typical distance of ~30AUwould subtend an angle of only 0.014arcsec. It is therefore possible to investigate their surface markings, shapes and rotational properties only through variations in the light that they reflect (their light curves). Here we report a survey of optical light curves from Kuiper-belt objects. Variations are observed only for the faintest objects in the survey. We can rule out eclipsing binary objects and variations in the surface markings as the origin of these light curves, suggesting that the observed variations are due to the rotation of irregularly shaped objects. Irregular shapes may be limited to the smallest Kuiper-belt objects because the material strength in their inner regions is sufficient to maintain the shape against the weight of the overlying material. If, however, all of the objects in our survey are of essentially the same size, then the intrinsically faintest ones may be composed of a stronger and darker material than the brighter ones.