A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

THE USE OF ONE IMAGINARY NUMBER AS PART OF THE RIGHT DIAGONAL (Part IIC)

Production of New Tables

This page continues from the previous Part IC. The purpose is to expand the first two tuples of Table XI (even number 288 and Table XII (odd number 289) and show that the tuples (±i,408,577) generates a different b for each table as well as different number of subtuples generated from the expansion of both tuples.

We take the first four lines of each table as shown below:

Table XI (Even Number 288)
δ₁i	ai	b	c	δ₂
	-288i	0	288
289i				289
	i	408	577

Table XII (Odd Number 289)
δ₁i	ai	b	c	δ₂
	-289i	0	289
288i				288
	-i	408	577

In order to keep the tables small enough we expand these two section of both tables to produce the following subtables where the even number n is incremented/decremented using the odd numbers:

Table XI_ex (Even Number 288)
δ₁i	ai	b	c	δ₂
	-288i	0	288
1i				1
	-287i	24	289
3	i			3
	-284i	48	292
5i				5
	-279i	72	297
7i				7
	-272i	96	304
9i				9
	-263i	120	313
11i				11
	-252i	144	324
13i				13
	-239i	168	337
15i				15
	-224i	192	352
17i				17
	-207i	216	369

Table XI_ex (Even Number 288)
δ₁i	ai	b	c	δ₂
19i				19
	-188i	240	388
21i				21
	-167i	264	409
23i				23
	-144i	288	432
25i				25
	-119i	312	457
27i				27
	-92i	336	484
29i				29
	-63i	360	513
31i				31
	-32i	384	544
33i				33
	i	408	577

and the odd number n is incremented/decremented using even numbers:

Table XII_ex (Odd Number 289)
δ₁i	ai	b	c	δ₂
	-289i	0	289
2i				2
	-287i	34	291
6i				6
	-281i	68	297
10i				10
	-271i	102	307
14i				14
	-257i	136	321
18i				18
	-239i	170	339
22i				22
	-217i	204	361

Table XII_ex (Odd Number 289)
δ₁i	ai	b	c	δ₂
26i				26
	-191i	238	387
30i				30
	-161i	272	417
34i				34
	-127i	306	451
38i				38
	-89i	340	489
42i				42
	-47i	374	531
6i				46
	-i	408	577

The final tuple for each is either (+i,408,577) or (-i,408,577) both of which lead to (-1, 166464, 332929) upon squaring. Note that the central number in the tuple, b, is incremented by 24 in the first case and 34 in the second. In addition, all the tuples generate legitimate right diagonals upon squaring even the ones that are factorable.

This concludes Part IIC. Go to either Part IIIC or Part ID to continue on tables of allowed tuples.

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