July 2018 little math problem

[Albert Chan]

Slight changes to my posted Python code (post 23) dramatically reduce checks down to 17 cases

Only patch is to change the loop range:
s range = [13, 14]
c range = range(max(1, ceg-8-9), min(9, ceg//2))
g range = range(c+1, min(10, 10 - s + ceg))

s range is reduced by taking advantage of 10-Complement Symmetry

c range is reduce because of this: c + e + g = ceg
min(c) = ceg - max(e + g) = ceg - 8 - 9
max(c) = ceg - min(e) - min(g) = ceg - 1 - (max(c) + 1)
max(c) * 2 = ceg - 2
max(c) < floor(ceg / 2)

g range is reduced because d = s - c - e = (s - ceg) + g
Since g is increasing inside loop, d is also increasing, remove invalid d, we get:
max(d) = (s - ceg) + max(g) < 10
max(g) < 10 - s + ceg

With above ranges, these are the only cases checked:

#1: 1 8 4 7 2 => [3, 9] 1 8 4 7 2 [5, 6]
#2: 1 9 3 7 3
#3: 2 9 2 8 3
#4: 1 5 8 4 2 => [6, 7] 1 5 8 4 2 [3, 9]
#5: 1 6 7 4 3 => [5, 8] 1 6 7 4 3 [2, 9]
#6: 1 7 6 4 4
#7: 1 8 5 4 5
#8: 1 9 4 4 6
#9: 2 6 6 5 3
#10: 2 7 5 5 4
#11: 2 8 4 5 5
#12: 2 9 3 5 6 => [4, 8] 2 9 3 5 6 [1, 7]
#13: 3 7 4 6 4
#14: 3 8 3 6 5
#15: 3 9 2 6 6
#16: 4 8 2 7 5 => [1, 9] 4 8 2 7 5 [3, 6]
#17: 4 9 1 7 6 => [2, 8] 4 9 1 7 6 [3, 5]

16X Symmetries: head swap, tail swap, reversing digits, 10-complement

Solutions = 6 primary solutions * 16 = 96