Wednesday, August 18, 2010

SEND MORE MONEY

One time in college I emailed my dad asking for money and he said I could have some if I solved the following equation:

   SEND
+MORE
_______
MONEY

Each letter represents its own digit (0-9) and multiple occurrences of the same letter represent the same digit (eg if one of the E's represents a 3, they all do).





Solution:
   9567
+ 1085
-----------
10652

17 comments:

  1. After several hours of vigorous problem solving and guess-and-check, I think I've come upon the solution.
    9567
    + 1085
    10652
    I actually think I've seen this exact problem in a MathCounts competition before...strange!

    ReplyDelete
  2. Replies
    1. HEy chralie Guthrie can you tell me in a easy way i couldn't get this fully.

      Delete
  3. Hi :)

    Actually I find exactly 155 solutions. Like so :

    9000 + 1000 = 10000
    9010 + 1090 = 10100
    9110 + 1001 = 10111
    9120 + 1091 = 10211
    9220 + 1002 = 10222
    9230 + 1092 = 10322
    9330 + 1003 = 10333
    9340 + 1093 = 10433
    9440 + 1004 = 10444
    9450 + 1094 = 10544
    9550 + 1005 = 10555
    9560 + 1095 = 10655
    9660 + 1006 = 10666
    9670 + 1096 = 10766
    9770 + 1007 = 10777
    9780 + 1097 = 10877
    9880 + 1008 = 10888
    9890 + 1098 = 10988
    9900 + 1199 = 11099
    9990 + 1009 = 10999
    9001 + 1000 = 10001
    9011 + 1090 = 10101
    9111 + 1001 = 10112
    9121 + 1091 = 10212
    9221 + 1002 = 10223
    9231 + 1092 = 10323
    9331 + 1003 = 10334
    9341 + 1093 = 10434
    9441 + 1004 = 10445
    9451 + 1094 = 10545
    9551 + 1005 = 10556
    9561 + 1095 = 10656
    9661 + 1006 = 10667
    9671 + 1096 = 10767
    9771 + 1007 = 10778
    9781 + 1097 = 10878
    9881 + 1008 = 10889
    9891 + 1098 = 10989
    9901 + 1189 = 11090
    9002 + 1000 = 10002
    9012 + 1090 = 10102
    9112 + 1001 = 10113
    9122 + 1091 = 10213
    9222 + 1002 = 10224
    9232 + 1092 = 10324
    9332 + 1003 = 10335
    9342 + 1093 = 10435
    9442 + 1004 = 10446
    9452 + 1094 = 10546
    9552 + 1005 = 10557
    9562 + 1095 = 10657
    9662 + 1006 = 10668
    9672 + 1096 = 10768
    9772 + 1007 = 10779
    9782 + 1097 = 10879
    9892 + 1088 = 10980
    9902 + 1189 = 11091
    9003 + 1000 = 10003
    9013 + 1090 = 10103
    9113 + 1001 = 10114
    9123 + 1091 = 10214
    9223 + 1002 = 10225
    9233 + 1092 = 10325
    9333 + 1003 = 10336
    9343 + 1093 = 10436
    9443 + 1004 = 10447
    9453 + 1094 = 10547
    9553 + 1005 = 10558
    9563 + 1095 = 10658
    9663 + 1006 = 10669
    9673 + 1096 = 10769
    9783 + 1087 = 10870
    9893 + 1088 = 10981
    9903 + 1189 = 11092
    9004 + 1000 = 10004
    9014 + 1090 = 10104
    9114 + 1001 = 10115
    9124 + 1091 = 10215
    9224 + 1002 = 10226
    9234 + 1092 = 10326
    9334 + 1003 = 10337
    9344 + 1093 = 10437
    9444 + 1004 = 10448
    9454 + 1094 = 10548
    9554 + 1005 = 10559
    9564 + 1095 = 10659
    9674 + 1086 = 10760
    9784 + 1087 = 10871
    9894 + 1088 = 10982
    9904 + 1189 = 11093
    9005 + 1000 = 10005
    9015 + 1090 = 10105
    9115 + 1001 = 10116
    9125 + 1091 = 10216
    9225 + 1002 = 10227
    9235 + 1092 = 10327
    9335 + 1003 = 10338
    9345 + 1093 = 10438
    9445 + 1004 = 10449
    9455 + 1094 = 10549
    9565 + 1085 = 10650
    9675 + 1086 = 10761
    9785 + 1087 = 10872
    9895 + 1088 = 10983
    9905 + 1189 = 11094
    9006 + 1000 = 10006
    9016 + 1090 = 10106
    9116 + 1001 = 10117
    9126 + 1091 = 10217
    9226 + 1002 = 10228
    9236 + 1092 = 10328
    9336 + 1003 = 10339
    9346 + 1093 = 10439
    9456 + 1084 = 10540
    9566 + 1085 = 10651
    9676 + 1086 = 10762
    9786 + 1087 = 10873
    9896 + 1088 = 10984
    9906 + 1189 = 11095
    9007 + 1000 = 10007
    9017 + 1090 = 10107
    9117 + 1001 = 10118
    9127 + 1091 = 10218
    9227 + 1002 = 10229
    9237 + 1092 = 10329
    9347 + 1083 = 10430
    9457 + 1084 = 10541
    9567 + 1085 = 10652
    9677 + 1086 = 10763
    9787 + 1087 = 10874
    9897 + 1088 = 10985
    9907 + 1189 = 11096
    9008 + 1000 = 10008
    9018 + 1090 = 10108
    9118 + 1001 = 10119
    9128 + 1091 = 10219
    9238 + 1082 = 10320
    9348 + 1083 = 10431
    9458 + 1084 = 10542
    9568 + 1085 = 10653
    9678 + 1086 = 10764
    9788 + 1087 = 10875
    9898 + 1088 = 10986
    9908 + 1189 = 11097
    9009 + 1000 = 10009
    9019 + 1090 = 10109
    9129 + 1081 = 10210
    9239 + 1082 = 10321
    9349 + 1083 = 10432
    9459 + 1084 = 10543
    9569 + 1085 = 10654
    9679 + 1086 = 10765
    9789 + 1087 = 10876
    9899 + 1088 = 10987
    9909 + 1189 = 11098

    You could have sent a different answer every week to your father, and get rich.

    ReplyDelete
  4. Or if every letter has to be a different digit, then you still have 21 solutions ;)

    9342 + 1093 = 10435
    9452 + 1094 = 10546
    9562 + 1095 = 10657
    9672 + 1096 = 10768
    9782 + 1097 = 10879
    9453 + 1094 = 10547
    9563 + 1095 = 10658
    9673 + 1096 = 10769
    9234 + 1092 = 10326
    9564 + 1095 = 10659
    9235 + 1092 = 10327
    9345 + 1093 = 10438
    9785 + 1087 = 10872
    9236 + 1092 = 10328
    9346 + 1093 = 10439
    9786 + 1087 = 10873
    9237 + 1092 = 10329
    9567 + 1085 = 10652
    9458 + 1084 = 10542
    9568 + 1085 = 10653
    9678 + 1086 = 10764

    Half a year worth of pocket money...

    ReplyDelete
  5. Ooops... sorry, the previous comment is obviously wrong... there's only one solution with all the digits being different, and that's the one you gave. My mistake, humbly !

    ReplyDelete
  6. 9235 + 1092 = 10327

    ReplyDelete
  7. It is the Constraint Satisfaction Method (CSM) from the Science of Artificial Intelligence.

    ReplyDelete
  8. Can you teach us how to solve it? I need to show my work for a homework assignment and im confused...?

    ReplyDelete
  9. I'm not sure how Matt did it, but I did a plug and chug where I started with some assumptions for S and M in order to get an answer in the 10,000s. You can then slowly see the relationship of what the numbers would have to be, again with a little plug and chug. Actually, I tried a value for M as zero and found an answer of 3712 + 0467 = 04179

    ReplyDelete
    Replies
    1. Interestingly my sister came up with the same result using M=0 before finding the result with M=1.

      I argued that M=0 is invalid due to 0 being a placeholder and irrelevant in that slot (where there are theoretically infinite zeroes.) But it is an applause-worthy unconventional solution, and it does work!

      Delete
  10. i have
    2817+368=3185
    2819+368=3187
    3719+457=4176
    3712+467=4179
    3829+458=4287
    3821+468+4289
    5731+647=6378
    5732+647=6379
    5849+638=6487
    6419+724=7143
    6415+734=7149
    6524+735=7259
    6853+728=7581
    6851+738=7589
    7316+823=8139
    7429+814=8243
    7539+815=8354
    7531+825=8356
    7534+825=8359
    7649+816=8465
    7643+826=8469
    8432+914=9347
    8542+915=9457
    and the last
    9567+1085=10652
    that's all.there is no other

    thanks.bye

    ReplyDelete
  11. Two four-digit numbers cannot sum to more than 20,000 so M=1.

    Now this sum must also be smaller than 12,000 so O=0.

    Looking at the hundreds column, it must include a carry and N=E+1. There is no carry in the thousands column so S=9.

    The sum in the tens column must satisfy N=E+1 and so R=8 and there is a carry.

    The only sum in the units column that causes a carry and uses the remaining digits is D=7, E=5, Y=2.

    So the solution is 9567+1085=10652.

    ReplyDelete
    Replies
    1. why must it be smaller than 12,000?

      Delete