Rewrite this so that we have positive operations on either side, because all the numbers to be used are positives:
a+cd=11+e/f+b+g. From this we can see that b and g are interchangeable, also that f must be a factor of e.
So e/f=4/2=2, 6/2=3, 8/2=4, 22/2=11, 8/4=2, 22/11=2. Right-hand side (R) minimum is 11+22/11+2+4=19. Maximum is 11+6/2+22+11+8=55. So 19≤R≤55 and similarly 13≤L≤250.
The aim is to get R and L to the same value without repetition of numbers on either side. Because R has the smaller range we need to find out all possible values for R, and we start by grouping according to e/f=2, 3, 4 and 11.
e/f |
b+g |
R |
L |
4/2=2 |
5+6=11 |
24 |
110, 187, 250 |
|
5+8=13 |
26 |
88, 143, 248 |
|
6+8=14 |
27 |
77, 121, 247 |
|
5+11=16 |
29 |
60, 140, 182 |
|
6+11=17 |
30 |
62, 118, 181 |
|
8+11=19 |
32 |
52, 116, 137 |
|
5+22=27 |
40 |
59, 74, 94 |
|
6+22=28 |
41 |
51, 63, 93 |
|
8+22=30 |
43 |
41, 61, 71 |
|
11+22=33 |
46 |
38, 46, 53 |
The first row that has a match of R and L is in bold and so we have a solution, e=4, f=2, b+g=11+22=33, a=6, cd=5*8=40. Note that b and g are interchangeable, as are c and d. There may be other solutions.
One solution out of several: a=6, b=11, c=5, d=8, e=4, f=2, g=22.
CHECK: 6-11+5*8-4/2-22=-5+40-2-22=40-29=11.