  This paper deals with the existence, or absence, of coalescent contractions
of contractible spaces. These are the contractions such that when the tracks of
any two points meet, at time t0, they remain together thereafter. If a finite
simplicial complex K is collapsible, then any collapse of K encodes coalescent
contractions of K. Examples of contractible spaces where no coalescent
contractions exist are the Dunce Hat and Bing's house. We establish a criteria
for contractible finite simplicial complexes that ensures there are no
coalescent contractions: the star-disc property. Keywords: contractible spaces,
coalescent homotopies, dunce hat, Bing's house.
