Hi Wei-Hwa, I'm now preparing a small booklet of MathDice problems for my G4G8 handout and will be checking my solutions with your program. I'm still learning the notation but will be asking about redundant solutions from your program. My solutions exclude roots so I won't be asking about that feature. The first one I may have found is solutions 5 and 7 below. best, dick 25 (8 ways) = 0.2 0.3 0.8 neg sum2+ root = 0.2 8 3 neg root neg root = 0.2 8 3 root neg pow = 0.32 8 prod_dn_x = 3 8 neg sum2+ 0.2 prod_nd_x neg = 3 neg 28 sum2+ = 3 neg 8 sum2+ 0.2 prod_nd_x = 3 neg 8 sum2+ 2 pow Hi again, I count the last two ways here as identical. Best, Dick 83 (3 ways) = 0.4 6 prod_dn_x neg 98 sum2+ = 0.6 49.8 prod_dn_x = 6 498 prod_dn_x Hi Wei-Hwa, Your program continues to function amazingly. My hat's off to you. I've got several cases here to be considered. In A the placement of neg brings nothing new. In B two identical digits get counted twice. In C I consider (28+7)/.5 and (27+8)/.5 to be the same. In D their are equavalencies but, more importantly, the following 3 cases were NOT FOUND by the program: 70=(2+5)/(.8-.7)=7/(.8-.5-.2)=7/(.5-.8/2). I did have trouble with inputting 1234. The file seems empty. Happy programming, Dick A: 24 (2 ways) = 0.7 4 neg sum2+ 7 prod_nn_x neg 0.9 sum2+ = 0.7 neg 4 sum2+ 7 prod_nn_x 0.9 sum2+ B: 128 (4 ways) = 0.8 6 prod_dn_x neg 8 sum2+ 7 neg pow = 0.8 8 prod_nd_x neg 0.6 sum2+ 7 neg pow = 8 8 prod_nn_x 6 7 prod_dn_x pow = 8 8 prod_nn_x 6 7 prod_dn_x pow C: 70 (11 ways) = 0.8 2 prod_nd_x 0.5 neg sum2+ 7 prod_dn_x neg = 0.8 2 prod_nd_x neg 0.5 sum2+ 7 prod_dn_x = 2 7 pow 58 neg sum2+ = 2 8 sum2+ 0.5 root 0.7 prod_nn_x = 5 neg 7 neg 82 sum3+ = 7 28 sum2+ 0.5 prod_nd_x = 7 8 sum2+ 0.2 prod_nd_x 5 neg sum2+ = 7 neg 28 neg sum2+ 0.5 prod_nd_x neg = 7 neg 8 neg sum2+ 0.2 prod_nd_x neg 5 neg sum2+ = 8 27 sum2+ 0.5 prod_nd_x = 8 neg 27 neg sum2+ 0.5 prod_nd_x neg D: 70 (11 ways) = 0.8 2 prod_nd_x 0.5 neg sum2+ 7 prod_dn_x neg = 0.8 2 prod_nd_x neg 0.5 sum2+ 7 prod_dn_x = 2 7 pow 58 neg sum2+ = 2 8 sum2+ 0.5 root 0.7 prod_nn_x = 5 neg 7 neg 82 sum3+ = 7 28 sum2+ 0.5 prod_nd_x = 7 8 sum2+ 0.2 prod_nd_x 5 neg sum2+ = 7 neg 28 neg sum2+ 0.5 prod_nd_x neg = 7 neg 8 neg sum2+ 0.2 prod_nd_x neg 5 neg sum2+ = 8 27 sum2+ 0.5 prod_nd_x = 8 neg 27 neg sum2+ 0.5 prod_nd_x neg Hi Wei-Hwa, As I use the program more I'll report other cases missed. Best, Dick Cases 1234 and 1445 gave empty solution files. In K missed cases are 85=1^(-2)-3*5=(2-.3)*5/.1=5*(2/.1-3)=(2^3+.5)/.1=(3^2-.5)/.1=2^3/.1+5=3^2/.1 -5. In L there are exactly duplicated solutions. K: 85 (3 ways) = 0.2 0.3 5.1 prod_ddn_x = 0.2 3 51 prod_ddn_x = 0.3 2 51 prod_ddn_x L: 35 (33 ways) = 0.2 0.4 0.6 sum2+ pow 7 prod_dn_x = 0.2 0.4 0.6 sum2+ root 7 prod_dn_x = 0.2 0.4 neg 0.6 neg sum2+ pow 7 prod_nn_x = 0.2 0.4 neg 0.6 neg sum2+ root 7 prod_nn_x = 0.2 0.6 sum2+ 4 7 prod_dnn_x = 0.2 4 prod_nn_x 0.6 neg sum2+ 7 prod_dn_x = 0.2 4 prod_nn_x neg 0.6 sum2+ 7 prod_dn_x neg = 0.2 7 prod_dn_x 0.4 0.6 sum2+ pow = 0.2 7 prod_dn_x 0.4 0.6 sum2+ pow = 0.2 7 prod_dn_x 0.4 0.6 sum2+ root = 0.2 7 prod_dn_x 0.4 0.6 sum2+ root = 0.2 7 prod_nd_x 0.4 neg 0.6 neg sum2+ pow = 0.2 7 prod_nd_x 0.4 neg 0.6 neg sum2+ pow = 0.2 7 prod_nd_x 0.4 neg 0.6 neg sum2+ root = 0.2 7 prod_nd_x 0.4 neg 0.6 neg sum2+ root = 0.2 neg 0.6 neg sum2+ 4 7 prod_dnn_x neg = 0.4 0.6 sum2+ 0.2 7 prod_ddn_x Hi Wei-Hwa, I've continued to find items you should be aware of and possibly use in future debugging efforts . In E expressions not found are 64=6/.1+4=(.1+.4)^(-6). In F expression not found is 16=.5^(-2^2). In G expression not found is 20=(5-3)/.1. In H expressions not found are 8=1/.1-2=(1-.2)/.1=.1^(-1)-2. In I expression not found is 5=(.3-.1)^(-2). In J expression not found is 8=(.1+.4)^(-3). I wanted to also bring up a gray area on equivalence. We know (a-b)^c = (b-a)^c if c is even and -(b-a)^c if c is odd. Should (7-5)^3 be equvalent to -(5-7)^3? I have chosen to treat them as equivalent. What would you say? Happy puzzling, Dick E: 64 (5 ways) = 1 64 prod_dn_x = 1 64 prod_nn_x = 4 16 prod_nn_x = 64 1 pow = 64 1 root F: 16 (16 ways) = 0.25 2 neg pow = 0.5 0.2 neg root 2 prod_nd_x = 0.5 2 2 neg pow neg root = 0.5 2 2 prod_nn_x neg pow = 0.5 2 neg 2 neg sum2+ pow = 0.5 2 prod_dn_x 2 pow = 2 0.2 root 0.5 prod_nn_x = 2 0.25 root = 2 0.5 2 neg pow pow = 2 0.5 2 pow root = 2 0.5 2 prod_dn_x pow = 2 2 0.5 neg root root = 2 2 0.5 root pow = 2 2 prod_nn_x 0.5 root = 2 2 sum2+ 0.5 root = 2 5 pow 2 prod_nd_x G: 20 (5 ways) = 0.15 3 prod_dn_x = 1 3 sum2+ 5 prod_nn_x = 1 5 sum2+ 0.3 prod_nd_x = 1 neg 3 neg sum2+ 5 prod_nn_x neg = 1 neg 5 neg sum2+ 0.3 prod_nd_x neg H: no solutions given for 8=f(112) I: 25 (2 ways) = 0.12 3 prod_dn_x = 0.2 1 3 neg sum2+ pow J: 8 (3 ways) = 1 3 4 sum3+ = 1 3 neg sum2+ 4 prod_nn_x neg = 1 neg 3 sum2+ 4 prod_nn_x