**Euler's Formulas for 3x3 Magic Squares of 6 Squares**

Euler wrote several papers about magic squares. He also derived several generating formulas for 3x3 magic squares of 6 squares (although in a disguised form). The generating formulas produce some, but not all solutions.

For each solution, there are several ways of putting them into a magic square and then testing for a 7th square.

You can potentially find 100-digit solutions with these formulas in a reasonable amount of time.

E806 contains fragments from Euler's notebooks. Sections 58, 59, and 60 have formulas that produce two 3-square arithmetic progressions having the same step value.

pp + qq = 2yy

rr + ss = 2xx

pp - qq
= rr - ss

Rearrange the entries of a 3x3 magic square to expose the arithmetic progressions.

c+a c-a-b c+b c-a-b
c-b c-b+a

c-a+b c c-b+a ---->
c-a c c+a

c-b
c+a+b c-a c-a+b
c+b c+a+b

The three rows, the three columns, and the two diagonals are all 3-square arithmetic progressions. Note that the row AP's all have the same step value.

We can put two 3-square arithmetic progressions having the same step value in three different row combinations.

Thus there are 9 different values to test for a 7th square for each of Euler's generated 6-square solutions.

pp yy qq pp
yy qq -- -- --

rr
xx ss -- -- -- pp
yy qq

-- -- -- rr
xx ss rr xx ss

The other 9 entries can be computed as follows.

(2rr-pp) (rr+pp)/2
(2pp-rr)

(2xx-yy) (xx+yy)/2
(2yy-xx)

(2ss-qq) (ss+qq)/2
(2qq-ss)

E796 contains generating formulas for the problem of finding three squares such that the sum of each two equals twice a square. Thus we need to find 6 squares that satisfy the following conditions.

pp + qq = 2zz

pp + rr = 2yy

qq + rr
= 2xx

There are three ways of assigning the six squares to the entries of the magic square. Thus there are 9 different values to test for a 7th square for each of Euler's generated 6-square solutions.

c-a-b c-b c-b+a

c-a c
c+a

c-a+b c+b c+a+b

pp zz qq qq
xx rr rr yy pp

yy
xx -- zz yy -- xx
zz --

rr -- -- pp
-- -- qq -- --

The other 9 entries can be computed as follows.

(xx-zz+qq) (yy-xx+rr)
(qq-xx+zz)

(rr-yy+xx) (pp-zz+yy)
(xx-rr+pp)

(rr-pp+qq) (pp-qq+rr)
(qq-rr+pp)

Rotating and reflecting the configuration does not produce any new entry values.

E797 concerns finding four values A,B,C,D satisfying the following conditions.

A+B = pp

A+C = qq

A+D = B+C = rr

B+D
= ss

C+D = tt

A < B < C < D

This is equivalent to finding a 3x3 magic square having 5 square entries as follows.

c+a c-a-b c+b C+D
2A B+D tt -- ss

c-a+b
c c-b+a ---> 2B A+D 2C
---> -- rr --

c-b c+a+b c-a A+C
2D A+B qq -- pp

You can safely ignore this paper because my 5-square solution is better and can be used in any arrangement to produce any 7-square solution.

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