Here, sd(s) means sum of digits of s.

My formula is as following: For problems of the form a->p->q->r->b with known variables a, b and unknown variables p, q, r; if there is an answer then it is p=sd(a)+sd(a+b), q=a+b, r=sd(b)+sd(a+b). Here is the proof: a=sd(p) (1), p=sd(a)+sd(q) (2), q=sd(p)+sd(r) (3), r=sd(q)+sd(b) (4), b=sd(r) (5) Since (1), (5), and (3), q=a+b. Now from (2) and (4) if you replace q with a+b, you get the formula above. This means that there can be up to 1 answer to problems of the form a->p->q->r->b with a, b known variables and p, q, r unknown variables.

I am wondering if there are similar formulas to other problems. Please share your explorations!

- aj