Szekeres and Turan found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows easily from Cauchy-Binet identity.) Later Turan [published][1] a simpler proof for the sum of the fourth powers but in Chinese. I vaguely remember that there are simpler probabilistic proofs for both cases. 

My question is about simple proofs for these identities, especially the one for 4th powers. 

Is there a formula for the 6th power? 


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Turan&s5=&s6=&s7=chinese&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=10&mx-pid=73555