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Constructing N

**Shian-Shyong Tseng, Bertrand M. T. Lin ^{*} and Sue Huei Liu^{#}**

National Chiao-Tung University

Hsinchu, Taiwan 300, R.O.C.

Ming-Chuan University

Taipei 111, R.O.C.

National Chiao-Tung University

Hsinchu, Taiwan 300, R.O.C.

An N_{£^} latin square of order *n* is an *n*¡Ñ*n* latin square containing no latin subsquare of order£^ for 1 <£^ < *n*. It has been shown in the literature that if *n*¡Ú2^{p}3^{q} there exists an *n*¡Ñ*n* latin square without latin subsquare of order£^ for£^< *n*. In this paper, combining with the known results, we show that for any integer n there is an *n*¡Ñ*n* N_{£^ } latin square if£^ is not a power of two .

Keywords: latin square, subsquare free, constructive proof

Received December 4, 1995; revised October 30, 1996.

Communicated by Wen-Lien Hsu.