E+E = N or N+10
If it's N, then G+H = E or E+10
If it's E, then D+C =D or D+10 (which it can't be, without C being 10)
So, if it's D, then C must be 0 (unlikely, as otherwise CACHE would begin with a zero)
and I+A = D, or D+10
If it's D, then R+C= I+10 If it's D+10, then R+C+1=I +10 (we know it's +10, because B and H can't be the same, so H= B+1)
If E+E=N+10
Then G+H+1=E or E+10
If it's E, then D+C=D, or D+10. (Making C a zero, unlikely, or 10, impossible)
If it's E+10, then D+C+1=D, or D+10 (Can't be D, as C would have to be -1, so C+1 must be 10, making C a 9)
So far, we know that H=B+1 and C is most likely a 9.
Now we know that I+A+1 = D or D+10
If it's D. then R+C=I+10
If it's D+10, then R+C+1=I+10, or R+C=I+9
As C=9, then R+C=I+10 can be written as R+9=I +10, or R = I+1
And R+C=I+9, would be R+9=I+9 (impossible, as R and I are different numbers)
So, we now know that R is 1 bigger than I
R=I+1
H=B+1
C=9
E+E=N+10, so N is 0,2,4,6 (can't be 8, as that would make E a 9)
That would mean that E is either 5,6,7 or 8
Trying the first option (N=0) would make E=5
G+H+1=5+10=15, so G+H=14, so 8+6, or 6+8, making B a 5 or 7.
As E=5, then B would be a 7
So, if N=0, then E=5, B=7, H=8, G=6 and C=9
I+A+1=D, so D must be 4 and I and A must add up to 3, making them 1 and 2.
We know that R=I+1, and only the 3 is left, so I must be the 2 and R would be the 3.
So, the order could be N A I R D E G B H and C
And, as it works out, I'm not going to check for any other N values.