By Ralph P. Grimaldi

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To appear. [11] C. D. Godsil and W. J. Martin, Quotients of association schemes, J. Combinatorial Theory, Series A, to appear. [12] J. Koolen, Euclidean Representations and Substructures of DistanceRegular Graphs. Ph. D. Thesis, Technical University Eindhoven, 1994. [13] C. Licata and D. L. Powers, A surprising property of some regular polytopes, Scientia, 1 (1988), 73-80. [14] W. J. Martin, Completely Regular Subsets. Ph. D. Thesis, University of Waterloo, 1992. [15] W. J. Martin, Completely regular designs of strength one, J.

The large sets of v + 1 MAD STS(v) constructed in Section 2 no longer have this nice property: a triple may occur once, twice or three times in such a set. On the other hand, in the large sets of v - 1 nearly disjoint STS(v) constructed in Section 3, each triple occurs once or twice, and again it is easy to see that the triples occurring twice form an STS(v). S. Griggs & A. Rosa 36 below also has this property. Each triple occurs once or twice, the latter forming a twofold triple system, TTS(13).

We call this bisecting. 3). , (e )) . 2 We will prove the result of Pelc (1987) that 25 questions suffice to find one of 220 objects in the 1-lie game. 1, bisecting (220,0) twenty times gives the state (1,20). The following questioning strategy shows that (1,20) is a winning 5-state. The obvious reductions of the winning 4-state (0,16) and of the state (0,4) have been omitted (these are states for a 0-lie game for which a simple bisecting strategy may be used). 5-state (1,20) [1,5] [0,15] N (0,16) (1,5) f 4-states [1,2] [0,3] (1,2) (0,4) 3-states [1,0] r [0,21 (1,0) 2-states (0,3) [0,2] [0,1] f (0,1) I/ % (0,2) (0,1) 1-states [0,11 [0,1] N (0,1) 0-states 46 R.