Octo Vertices (Posted on 2008-12-30)

In the following octahedral net the vertices A, B, C, D, E and F, and are to be assigned distinctive values from 1 through 9.


M, N, O, P, Q, R, S and T are the values of the respective sums of the three vertices which form the respective surfaces to which each is assigned.

M + N = O + P = Q + R = S + T

Find unique sets of values¹ for A, B, C, D, E and F such that the values of M through T (but not necessarily in that order) form series which increment by 2 of which there are 7.

Background statistics:
                                   Unique        Actual
                M+N=   Interval     Sets2       Solutions
                 21       1           1            96
                 23       1           2            96
                 25       1           2            96
                 26       2           2            96
                 27       1           5           480
                 29       1           4           288 
                 30       2           3           144
                 31       1           8           288
                 33       1           6           480
                 34       2           2            96
                 35       1           2            96
                 37       1           2            96
                 39       1           1            96
                                     40          2448 

Note:
1. For M+N=21,
          1, 2, 3, 6, 4, 5
          1, 2, 3, 6, 5, 4
      and 1, 2, 5, 3, 4, 6  
   are the first 3 of 96 solutions.  The 96 values do not
   discriminate amongst vertex rotation or reflections. 
2. Each unique set configures the octahedron in more ways than one.

	Submitted by brianjn
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Solution:	(Hide)
M+N A B C D E F 26 1 2 9 4 3 7 26 1 2 9 6 3 5 30 1 2 9 6 5 7 30 1 3 9 5 4 8 30 1 4 9 6 3 7 34 1 3 9 7 6 8 34 1 4 9 8 5 7 The following program generates all of the data noted in the background statistics summary. DECLARE SUB keyin () DECLARE SUB validate () DECLARE SUB order () DIM SHARED sum(10) DIM SHARED a, b, c, d, e, f DIM SHARED m, n, o, p, q, r, s, t DIM SHARED line1$, line2$, line3$, line4$ DIM SHARED interval DIM SHARED count count = 1 'line1$ = "" OPEN "octo.txt" FOR OUTPUT AS #1 CLS FOR a = 1 TO 9 FOR b = 1 TO 9 FOR c = 1 TO 9 FOR d = 1 TO 9 FOR e = 1 TO 9 FOR f = 1 TO 9 '----------------------------------------------- REM Isolate when a,b,c,d,e and f have unique values in relation to each other '----------------------------------------------- IF (e <> f AND d <> e AND d <> f AND c <> d AND c <> e AND c <> f AND b <> c AND b <> d AND b <> e AND b <> f AND a <> b AND a <> c AND a <> d AND a <> e AND a <> f) THEN line1$ = STR$(a) + STR$(b) + STR$(c) + STR$(d) + STR$(e) + STR$(f) + " " '-------------- REM Vertex sums '-------------- m = c + d + f: n = a + b + e: o = b + c + f: p = a + d + e q = a + b + f: r = c + d + e: s = a + d + f: t = b + c + e line2$ = line1$ + STR$(m) + STR$(n) + STR$(o) + STR$(p) + STR$(q) + STR$(r) + STR$(s) + STR$(t) + " " '-------------------------------- REM Map vertex sums to the array '-------------------------------- sum(0) = m: sum(1) = n: sum(2) = o: sum(3) = p sum(4) = q: sum(5) = r: sum(6) = s: sum(7) = t order 'REM Routine reorders the array in ascending values validate 'REM Routine only accepts and saves sequences which REM have A.P. intervals of either 1 or 2. END IF NEXT: NEXT: NEXT: NEXT: NEXT: NEXT CLOSE 1 SUB keyin DO LOOP WHILE INKEY$ <> CHR$(32) END SUB SUB order FOR i = 0 TO 6 FOR j = i + 1 TO 7 IF sum(i) > sum(j) THEN y = sum(i) sum(i) = sum(j) sum(j) = y END IF NEXT: NEXT line3$ = "" FOR x = 0 TO 7 line3$ = line3$ + STR$(sum(x)) NEXT END SUB SUB validate line4$ = line2$ + line3$ interval = 1 s = 0 '------------------------------- REM Test for A.P. interval of 1 '------------------------------- DO s = s + 1 IF sum(s + 1) - sum(s) <> 1 THEN interval = 0 END IF LOOP WHILE (interval = 1 AND s <= 5) '------------------------------- REM Test for A.P. interval of 2 '------------------------------- IF interval = 0 THEN s = 0 interval = 1 DO s = s + 1 IF sum(s + 1) - sum(s) <> 2 THEN interval = 0 END IF LOOP WHILE (interval = 1 AND s <= 5) END IF '------------------------ REM Save valid results '------------------------ IF interval = 1 THEN PRINT "Count"; count; m + n; "Line4$"; line4$ line4$ = STR$(count) + " " + STR$(m + n) + " " + line4$ PRINT #1, line4$ count = count + 1 ' keyin END IF END SUB

	Subject	Author	Date
	re: computer solution	brianjn	2008-12-31 22:23:10
	computer solution	Charlie	2008-12-30 17:45:22
	A wee start analytically	rod hines	2008-12-30 17:41:36