The Magic Encyclopedia ™

Generating Squares
(by Aale de Winkel)

recent discussion with <Gil Lamb> took me to investigate what Gil called Generating Squares, later Gil changed the definition to some subset. <George Chen> reckognized these squares as Candy's "Auxiliary Squares" and provided a recursive counting argument. squares that share with the "Normal order m square" N_m[x,y] = x + my the property that all (broken) diagonals sum up to the magic sum S_m = m(m²-1)/2 (note I use the analitic numberrange in this article). The monagonals can sum up to any number (but in incremental order) as they do in N_m.
Currently I believe that all "Generating Sqares" can be obtained as multiplication of "Normal Rectangles"
_pN_q[x,y] = x + py ; x = 0..p-1, y = 0..q-1
A product which can be put into one formula. The listing below gives Generating Squares in normalized position.
This gives a definite amount of Generating Squares based on the type of the order m.

Auxiliary Rectangles / Normal Rectangles
<George Chen> Number of Auxiliary rectangles: #(1Np) = 1; #(pN1) = 1 (? 0 ?) ; #(pNq ; q prime) = 1; #(pNq) = #(1Nq + r|p,r>1∑ [ s prime,s<q,q|n ∑ #(rNq/s) - t ∑ #(rNt) ] with t = GCD(q/q1,q/q2}, q1 , q2 two distinct prime factors of q.
Special cases	#(p2Np2) = 3; #(pnNpn) = (2n-1)!/[n!(n-1)!]; #(pN(tuqv)) = (u+1)(v+1)-1; #(pNq2) = #(prime factors of p); #(pqNrs) = 7;
Generating Squares
The following lists of Generating Squares generates them as products of "Normal Rectangles". These products can be stated in one general formula.
(i=0∏n piNqi)[x,y] = i=0∑n k=0∏i pkqk { ([x % k=0∏i pk] \ k=0∏i-1 pk) + pn ([y % k=0∏i qk] \ k=0∏i-1 qk) }; i=0∏n pi = i=0∏n qi = m; x,y = 0..m-1
Examples up till 4 rectangles: (pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [(x\r) + p (y\s)] (tNu * pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] + pqrs [(x\pr) + t (y\qs)] (vNw * tNu * pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] + pqrs [((x%prt)\pr) + t ((y%qsu)\qs)] + pqrstu [(x\prt + v (y\qsu)]
prime order	Nm
orders m = pq p,q prime	p,q	r = p
	Npq	Npq
	qN1 pNpq pN1 qNpq	pN1 pNpp
	pNp qNq qNq pNp qNp pNq pNq qNp	pNp pNp
orders m = pqr p,q,r prime note: '...' deactivated from display	p,q,r	r = p <> q	p = q = r
	Npqr	Nppq	Nppp
	pN1 qrNpqr qN1 prNpqr rN1 qrNpqr	pN1 pqNppq qN1 ppNppq	pN1 ppNppp
	pqN1 rNpqr qrN1 pNpqr prN1 qNpqr	ppN1 qNppq pqN1 pNppq	ppN1 pNppp
	...	pNq pqNpp qNq ppNpp pNp pqNpq qNp ppNpq	pNp ppNpp
	...	ppNq qNpp pqNq pNpp ppNp qNpq pqNp pNpq	ppNp pNpp
	...	pN1 pNq qNpp pN1 qNq pNpp qN1 pNq pNpp pN1 qNp qNpq qN1 pNp pNpq pN1 pNp pNpq	pN1 pNp pNpp
	...	pNpq pqNp qNpq ppNp pNpp pqNq qNpp ppNq	pNpp ppNp
	...	ppNpq qNp pqNpq pNp ppNpp qNq pqNpp pNq	ppNpp pNp
	...	pN1 pNpq qNp pN1 qNpq pNp qN1 pNpq pNp pN1 pNpp qNq pN1 qNpp pNq qN1 pNpp pNq	pN1 pNpp pNp
	...	pNq pNp qNp pNq qNp pNp qNq pNp pNp pNp pNq qNp pNp qNq pNp qNp pNq pNp pNp pNp qNq pNp qNp pNq qNp pNp pNq	pNp pNp pNp
orders m = pqrs p,q,r prime note: '...' not yet estimated	p,q,r,s	....	p = q = r = s
	Npqrs	....	Npppp
	.....	....	pN1 pppNpppp ppN1 ppNpppp pppN1 pNpppp
	.....	....	pNp pppNppp ppNp ppNppp pN1 pNp ppNppp pppNp pNppp pN1 ppNp pNppp ppN1 pNp pNppp
	.....	....	pNpp pppNpp ppNpp ppNpp pN1 pNpp ppNpp pNp pNp ppNpp pppNpp pNpp pN1 ppNpp pNpp ppN1 pNpp pNpp pNp ppNp pNpp ppNp pNp pNpp pN1 pNp pNp pNpp
	.....	....	pNppp pppNp ppNppp ppNp pN1 pNppp ppNp pNp pNpp ppNp pNpp pNp pNp pppNppp pNp pN1 ppNppp pNp ppN1 pNppp pNp pNp ppNpp pNp ppNp pNpp pNp pN1 pNp pNpp pNp pNpp ppNp pNp ppNpp pNp pNp pN1 pNpp pNp pNp pNp pNp pNp pNp

Note The above lists definite Generating Squares (GS) based on the type of the order, these kind of squares are rapidly available by multiplication of "Normal Rectangles" as above listed. This means that only N_m is reckognized as a "Auxiliary Square" when m is a prime order. For orders like 4, 9, and 25 only three GS's are obtained, expanding naturally to 7 GS's for other double prime orders (like 6,10 and 14), three primes enter with order 8 which has 10 GS's which corrrespond with 42 GS's for orders like 12, 18 and 20. The above systematic approach for four equal primes gives us 35 GS's for orders like 16 (This I haven't (yet?) estimated for the other quadruple prime orders)