Finding the Digits of SEND + MORE = MONEYDate: 05/19/98 at 23:34:59 From: Echo Subject: A Basic Story Problem Of Guessing Mystery Numbers Here is one of the more well-known cryptic problems, so I have been told, but I am stuck. I really don't know quite what to do or where to start. If this is one of the problems in your archives, please let me know. I looked, but was unable to find it. SEND + MORE ------ MONEY I am trying to find what each of the letters represents. Thanks so much for your help. Date: 05/20/98 at 13:04:05 From: Doctor Rob Subject: Re: A Basic Story Problem Of Guessing Mystery Numbers These problems can be reduced to systems of equations and inequalities: (1) D, E, M, N, O, R, S, Y different whole numbers (2) 0 <= D <= 9 (3) 0 <= E <= 9 (4) 1 <= M <= 9 (it's the leading digit of "MORE", so not zero) (5) 0 <= N <= 9 (6) 0 <= O <= 9 (7) 0 <= R <= 9 (8) 1 <= S <= 9 (it's the leading digit of "SEND", so not zero) (9) 0 <= Y <= 9 (10) D + E = Y + 10*a (units column, carry a) (11) N + R + a = E + 10*b (tens column, carry b) (12) E + O + b = N + 10*c (hundreds column, carry c) (13) S + M + c = O + 10*d (thousands column, carry d) (14) d = M (ten-thousands column) (15) 0 <= a <= 1 (16) 0 <= b <= 1 (17) 0 <= c <= 1 (18) 0 <= d <= 1 (19) a, b, c, d are whole numbers First of all, from (4), (14), and (18), 1 <= M = d <= 1, so: (18a) d = 1 (4a) M = 1 Now, in (13), S + 1 + c = O + 10. If c = 0, by (8), (13), and (6), 9 >= S = O + 9 >= 9 so S = 9 and O = 0. If c = 1, 9 >= S = O + 8 >= 8, so either S = 8 and O = 0, or S = 9 and O = 1. By (1) and (4a), O cannot be 1, so in either case: (6a) O = 0 (13a) S + c = 9 Looking at (12), (3), and (16): 9 = 8 + 1 >= E + b = N + 10*c > 10*c so c < 9/10, and by (17) and (19): (17a) c = 0 and by (13a): (8a) S = 9 From (12), (6a) and (17a), E + b = N, so, by (1): (16a) b = 1 (12a) E + 1 = N Now, from (11) and (16b), N + R + a = 10 + E. Putting this together with (12a): (11a) R + a = 9 which tells you by (1), (15), and (8a) that: (7a) R = 8 (15a) a = 1 From (10) and (15a): (10a) D + E = 10 + Y Now, consider the possibilities for Y. It must be 2, 3, 4, 5, 6, or 7. Use (10a) and (12a) along with (1), (4a), (6a), (7a), and (8a). So far we have: 1 0 1 1 9 E N D + 1 0 8 E ---------- 1 0 N E Y From here, you can finish by yourself. For more assistance, refer to this answer from the Dr. Math archives: Send More Money http://mathforum.org/library/drmath/view/57951.html -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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