TITLE: A simple proof for a theorem of Szekeres and Turán
QUESTION [11 upvotes]: Szekeres and Turán 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 Turán published 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?

REPLY [5 votes]: This is an answer, but my insight is hardly a mathematical one.  Nonetheless, here we go:
The 1955 paper of Turán is, in fact, published in both English and Chinese in the same journal.  The Chinese version is pages 411-417 in the journal; the English language version is pages 417 - 423.  
Turán writes: "I am pretty sure that on the way one can evaluate $M_n(6)$ or $S_n(6)$ after a little longer calculations and also further."  But he doesn't give a formula.