Here arre some equations that need Solving - Questions are

Can we Goal seek ?

The puzzle was built for Excel - with 24 variables.

How to maintain INTEGER SOLUTION ?

Use natural Letter column Heads for the side lengths

Solution is 38 - side length of Square N in my notation.

Text of Equations.  Essentially the idea is to set the lowest few values and let the rest build up..

Assume I,J

J or X are smallest

Logical order

First guess at some equations - see right for thoughts developing...

G = I + J

H = G + J

B = D + E + F

C = O + P

P = W + V

H = G + J

D + G + I

E = G + H

A = B + D

S = R + X

Q = R + S

E = G + H

D + G + I

N = K + L

R = T + X

F = E + H

G = I + J

M = K + N

W = U + V

1) Set X, then manually solve U

B = D + E + F

F = E + H

=

A = B + D

2) reverse solve the top left ABMN rectangle.

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Now need more properties of the ABMN rectangle

unknowns

L = F + E - I

55

39

81

16

9

14

4

5

3

1

18

20

56

38

30

51

64

31

33

29

35

8

43

2

Now we have sim eqn's

K = A + D + I - M

Which defines it now in terms of knowns except I

But is circular on M

And L is now N - K

But Better L is F + E - I

A+B = M + N

O = B + F + L + N - C

BUT O and C defined in terms of each other.

N = K + L

what about AMQ-TUC

PV-T ?

Ok

M = K + N

T = M + N + O - Q - R

Can perhaps make this into

SU - RV

UX-V ?

U = R + T + V - S

subs

A+B = K + 2 N

And we have 4 equations for Z

A+ B = K + 2K + 2L

Surely we just need to keep inserting all possible triplets for I,J,X until it solves.   BUT CANNOT FIND FORMULAES FOR UVK either

3K = A + B - 2 L

Seems that subtractive formulaes are like saying you have unknowns…

Aha and now need to find I, J that make K integer

Try a different tack

See Right scroll

2,1

So we sum of squares is Z square where Z is

5,1

AMQ

QSUW

WPC

ABC

8,1

175

175

175

175

7,2

30625

30625

30625

30625

10,2

QSUW

is directly adjustable

on U

to get the others closer together

etc…

Have to make th Z match on bottom and Right

Started with eyeballing the Diagram in Observer

J is smallest

Now if we examine Lower corner correctly

then X, I G

Assume X and U are unkown

V looks about G + H = E

S = U-X

U is huge…

R=S-X

T=R-X

3025

1521

6561

256

81

196

16

25

9

1

324

400

3136

1444

900

2601

4096

961

1089

841

1225

64

1849

4

V= U+X-T

and we solved V

30625

Curiously Guess X = 2 and play with U till QSUW = WPC finds U is 35

But X=1 then U doesn't solve,

16-17

Now maybe just need to tweak I, J until AM and AB get big enough

And X= 3

it is 52-53

But at sheet 5 attempt we see that with J=1 I=8 AMQ gets too big

X=4 => U= 70

and a pattern

Looks like getting near a method - for increasing X - find U then look for I,J to complete the size of top left.

Can we set up equations that Back solve the top rectangle ? - Yes - gaol seek it.

A+M = C + O = Z - Q

Z=length of outer side.

A + B  = Z - C

M+N = Q + R + T - O

Well yes we have it down to one unknown hence goal seek will find the others

Set A to satisfy…

z =

175

c=

81

q=

64

55

39

81

16

9

14

4

5

3

1

18

20

56

38

64

175

B=Z-C-A

1

0

M=Z-Q-A

D=A-B

N=A+B-M

K=M-n

L=n-k

f=A+M-b-n-l

e=b-d-f

h=f-e

Goalseek to zero

g=e-h

j=L-f-h

By changing A

Hmm also h-g

i=g-j

Fix some errors - look for consistency in fwd and Back

Obviously the solution is scaleable…

X increasing in multiples of 2