ultramagic squares of order 7

Ultramagic Squares of Order 7

Results

Partial tasks

The calculation was divided into 90 parts. In each part n(1,1) and n(2,1) were fixed.
T(1,x) is the total number of squares with n(1,1) = 1 and n(2,1) = x and n(1,2) > x
T(x,1) is the total number of squares with n(1,1) = x and n(2,1) = 1 and n(4,6) > x

Restrictions for the parameter x

The following values are not available for the parameter x:
1	because the number 1 can only occur once.
25	because n(4,4) = 25 is a precondition for the square to be self-complementary.
49	because n(6,7) = 49 or n(7,7) = 49 as 49 is the complement value of 1.
48	because in the case n(1,1) = 1 is x = n(2,1) < n(1,2) < 49 because in the case n(2,1) = 1 is x = n(1,1) < n(4,6) < 49

Table of partial results

x	T(1,x)	T(x,1)	x	T(1,x)	T(x,1)	x	T(1,x)	T(x,1)
2	10971	11032	17	54163	33714	33	56225	40940
3	14805	12388	18	54928	39271	34	54913	45353
4	17422	15318	19	58294	35674	35	53394	40684
5	21499	16172	20	58114	40675	36	49525	43648
6	24188	19825	21	60299	37195	37	45498	35885
7	28539	19968	22	59509	40657	38	43142	40704
8	30060	23778	23	60892	36256	39	40530	34942
9	34409	22698	24	58887	41212	40	35467	36479
10	36486	27437	26	68566	47458	41	31393	30900
11	40332	25558	27	69538	42359	42	27436	31202
12	41629	30730	28	66469	47145	43	23581	25281
13	45905	29409	29	64984	42210	44	19033	23581
14	46978	34453	30	61937	47938	45	14082	16880
15	49326	30872	31	63046	42199	46	9854	13013
16	51005	37091	32	58975	46927	47	4562	7213

Total result

a = T(1,2) + T(1,3) + ... + T(1,47)	b = T(2,1) + T(3,1) + ... + T(47,1)
a = 1,920,790	b = 1,444,324
There are c = a + b so-called essentially different squares, i.e. it is not possible to derive one square from another by one of the 48 transpositions.
c = a + b = 3,365,114
2a is the total number of squares with n(1,1) = 1, including reflections at the downward main diagonal, i.e. without the condition n(2,1) < n(1,2).	2b is the total number of squares with n(2,1) = 1, including K³-transposed squares, i.e. without the condition n(1,1) < n(4,6).
2a = 3,841,580	2b = 2,888,648
There are 24 transpositions that move the number from cell (1,1) to any of the 24 axis-cells. Thus you get 24 · 2·a = 48·a different squares with number 1 in any axis-cell.	There are 24 transpositions that move the number from cell (2,1) to any of the 24 filling-cells. Thus you get 24 · 2·b = 48·b different squares with number 1 in any filling-cell.
Finally there are e = 48·c squares including rotated and reflected squares. Each of these squares could be generated from one of the c essentially different squares by one of the 48 transpositions.
e = 48 · c = 161,525,472
As it is not usual to count reflected and rotated squares, the total number of different self-complementary pan-magic 7x7-squares is d = e / 8 .
d = e / 8 = 6 · c = 20,190,684
Notice that all 3,365,114 essentially different squares were saved on disk. All other squares could be derived very easily.

Reliability

Of course I can not garantee that the results are correct. But I'm rather sure they are.
- All calculated magic squares had been tested with regard to the demanded properties.
- There are no doublets within the calculated squares.
- It could be shown that there are 3456 regular pandiagonal symmetric magic 7x7-squares. My computer-program found exactly that amount of these special squares.
- In June, 2004 Francis Gaspalou from Gif-sur-Yvette, France told me about his own computations. He got the same total number of ultramagic order-7 squares. See improvements.

Link

Prof. Grogono describes the properties of regular pan-magic squares.

Question

Can anybody confirm one of the partial results in the above table?

Index

summary	cells	equations	transpositions	improvements
results	programs	files	conclusions	samples