1. Pólya's walk in the Euclidean plane is the motion of a particle, which takes successive independent random steps of unit length parallel to the coordinate axes, the probabilities of each of the four possible directions at each step being ¼. We imagine the particle starts at the origin, and that, in taking the ith step (i = 1,2,…) from (xi-1, yi-1) to (xi, yi), it leaves a straight track of unit length between these two points. Let Tn denote the track thus marked out in the first n steps; so that Tn is a connected chain of n straight segments each of unit length. Define the interior of Tn to be the set of points, which neither lie on Tn nor can be connected to the point at infinity without crossing Tn at least once; and let An denote the area of the interior of Tn. Associated with each walk ω, there is a sequence of random numbers {An(ω)}; what can be said about the behaviour of this sequence, especially for large n? We shall proveTheorem 1. For all ω,0 ≤ An(ω) ≤ 1/16 n2.