e consider the problem of comparing v treatments in b blocks of size k each, for experimental situations in which the responses are affected by the spatial or temporal position of the experimental units or plots within block. A common linear trend over plots within blocks is assumed.;We show that the conjecture is true in many situations, one of which is when k is even. We also prove that the conjecture is true for Balanced Incomplete Block (BIB) designs. For cases when the trend-free designs do not exist we obtain 'nearly' trend-free designs (definition due to Yeh, Bradley and Notz (1985)). Optimality properties of some of the trend-free and nearly trend-free designs are also investigated.;Finally, with the help of Hall's SDR algorithm (1956), we develop an algorithm and a fortran program to convert block designs to linear trend-free block designs.;In the above experimental situations, the treatments orders within blocks are important. The question is: how to arrange treatments into plots to get a highly efficient design? Trend-free designs, introduced by Yeh and Bradley (1980) have some very desirable statistical properties. Yeh and Bradley (1983) conjectured that each binary incomplete design in which every treatment replication is r, can be converted into a linear trend-free design by rearranging treatments into plots within blocks if and only if
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