Geometry: unique four-sided polygons

[Scrapyard]

Good Afternoon,

I'm trying to assign values (whole number > 0 but <= 5) to each side of a four-sided polygon (sides a, b, c, d) in which the values add up to the same total, such as 12. However, the polygon must be unique regardless of orientation.

One way I was visualizing this using Excel was to think of each row as a single polygon with Columns A, B, C, D as sides (making up a single array). However, I'm running into two basic problems:
- I am uncertain how to compare these arrays easily. If I use more columns to simulate a single rotation clockwise three additional times (columns E, F, G, H; I, J, K, L; M, N, O, P), I can compare across the four sets, looking for unique values. This seems like a very clunky way to accomplish this, and I haven't gotten in to work.
- Likewise, manually inputting in each value, hoping not to miss one, seems awfully arduous.

For example:

	A	B	C	D
1	3	3	3	3
2	3	3	2	4
3	3	3	4	2
4	4	2	3	3

Row 4 is not unique because it is the same as Row 3.

I appreciate any help anyone can offer in finding an easier way to accomplish this task (hopefully without VBA).

 

[Snakehips]

@Scrapyard Intriguing.
is the scale of this such that you are happy to enter values manually but would like an indication when a set is not unique?
You mention rotation in respect of orientation. Hence 3 and 4 are not unique.
What about 2, which is a flipped version of 3 and 4 ?

 

[Scrapyard]

@Scrapyard Intriguing.
is the scale of this such that you are happy to enter values manually but would like an indication when a set is not unique?
You mention rotation in respect of orientation. Hence 3 and 4 are not unique.
What about 2, which is a flipped version of 3 and 4 ?

Sorry if it wasn't clear...

Each four-sided polygon must be unique. The sides can be arranged differently as long as rotating it clockwise or counterclockwise does not produce the same result. I'm not sure about flopping or flipping.

Another way to visualize what I'm trying to do ... Consider the values are labels and not related to the size ... I'm attempting to cut out "squares" of paper (the top is white and bottom is black), put labeled numbers along the edge and centered on each side of the square (on the top/white) that add up to a given value (say 12), without any duplicates.

I am trying to see how these arrays are arranged and the total number of "squares" required.

---3---        ---3---        ---3---       ---4---
|     |        |         |        |         |       |         |
3     3       4        3       2        3      3   x   2
|     |        |         |        |         |       |         |
---3---        ---2---        ---4---       ---3---

 

[Scrapyard]

Sorry, I was trying to get the graphic done, but timed out.

 

[Snakehips]

Thanks. I will take a look later and see if I can make sense / offer any help.
Don't hold your breath!

 

[Snakehips]

@Scrapyard Ok, not sure if this helps or not?
Enter the Target in G1
Utilises helper columns H:M which you may hide.
Data validation limits Input in B:E to 1thro 5
Conditional formatting highlights any 'off target' row, in yellow.
Conditional formatting highlights any non-unique sets, in red.

Book2
	A	B	C	D	E	F	G	H	I	J	K	L	M
1	#	A	B	C	D	Target	12	Off Target	Not Unique	Actual	Var1	Var2	Var3
2	1	3	3	3	3			FALSE	FALSE	3333	3333	3333	3333
3	2	3	3	2	4			FALSE	FALSE	3324	3243	2433	4332
4	3	3	3	4	2			FALSE	TRUE	3342	3423	4233	2334
5	4	4	2	3	3			FALSE	TRUE	4233	2334	3342	3423
6	5							FALSE	FALSE
7	6	5	1	3	3			FALSE	FALSE	5133	1335	3351	3513
8	7	4	3	3	5			TRUE	FALSE	4335	3354	3543	5433
9	8	4	1	3	3			TRUE	TRUE	4133	1334	3341	3413
10	9	1	3	3	4			TRUE	TRUE	1334	3341	3413	4133
11	10							FALSE	FALSE
12	11	2	2	4	4			FALSE	FALSE	2244	2442	4422	4224
13	12	1	1	5	5			FALSE	TRUE	1155	1551	5511	5115
14	13	5	5	1	1			FALSE	TRUE	5511	5115	1155	1551
15	14	1	5	5	1			FALSE	TRUE	1551	5511	5115	1155
16	15							FALSE	FALSE
17	16							FALSE	FALSE
18	17							FALSE	FALSE
19	18							FALSE	FALSE
20	19							FALSE	FALSE
21	20							FALSE	FALSE
Sheet1

Cell Formulas
Range	Formula
H2:H21	H2	=AND(SUM(B2:E2)<>$G$1,SUM(B2:E2)>0)
I2:I21	I2	=AND(J2>"",IF(COUNTIF(J2:M2,J2)=4,COUNTIF($J$1:$J$21,$J2)>1,COUNTIF($J$1:$J$21,$J2)+COUNTIF($J$1:$J$21,$K2)+COUNTIF($J$1:$J$21,$L2)+COUNTIF($J$1:$J$21,$M2)>1))
J2:J20	J2	=B2&C2&D2&E2
K2:K20	K2	=C2&D2&E2&B2
L2:L20	L2	=D2&E2&B2&C2
M2:M20	M2	=E2&B2&C2&D2

Named Ranges
Name	Refers To	Cells
_FilterDatabase	=Sheet1!$E$4:$I$18	H4, J4:M4

Cells with Conditional Formatting
Cell	Condition		Cell Format	Stop If True
A2:M21	Expression	=$H2	text	YES
A2:M21	Expression	=$I2	text	NO

Cells with Data Validation
Cell	Allow	Criteria
B2:E21	Whole number	between 1 and 5

NB: Ignore the named range reference above. It is not relevant to this.