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Inheritance Envelopes (Posted on 2004-12-24) Difficulty: 3 of 5
A wealthy man had three sons all of whom were quite good at math and logic. To get a share of his inheritance each had to correctly determine a positive integer which he had chosen. He told them that the number had four different non-zero decimal digits, in ascending order.

He prepared three sealed envelopes each of which contained a number. The first contained the product of the four digits, the second contained the sum of the squares of the four digits, the third contained the sum of the product of the first two digits and the product of the last two digits, and the envelopes were clearly marked as such. He showed the three envelopes to the three sons and had them each take one at random. Each one saw the number inside his envelope but didn't see the number inside the other two envelopes.

The sons were stationed at three different computers so that they couldn't communicate with one another (but were linked to the father's computer). After one hour they could submit a number or decline. Anyone who submitted a wrong answer would be eliminated and get nothing. If one or more submitted the correct answer they would each receive a share of the inheritance, and the contest would end with the others getting nothing. If no one submitted the correct answer they would be instructed to work on the problem for another hour. The process would repeat as often as necessary. Each of the sons decided not to submit an answer unless they sure it was correct.

At the end of the first hour no one had submitted an answer. At the end of the second hour no one had submitted an answer. At the end of the third hour no one had submitted an answer. At the end of the fourth hour all three of them submitted the correct answer!

Can you determine the number?

See The Solution Submitted by Jim    
Rating: 3.5000 (4 votes)

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Solution re: computer solution--listing improvement | Comment 2 of 4 |
(In reply to computer solution by Charlie)

In the lists that follow, at each stage the numbers that are unique within the column in the current group are marked with an asterisk, indicating in the first three groups that that any line with an asterisk should be dropped in the next group, and in the last that if all three are marked with an asterisk, it is the answer

 126 
1234    24*   30*   14*
1235    30*   39*   17*
1236    36*   50*   20*
1237    42*   63*   23
1238    48    78    26
1239    54*   95    29*
1245    40*   46*   22*
1246    48    57*   26
1247    56*   70*   30
1248    64*   85*   34
1249    72   102    38
1256    60    66*   32*
1257    70*   79*   37*
1258    80*   94    42
1259    90   111    47
1267    84    90    44
1268    96   105    50
1269   108   122*   56
1278   112*  118*   58
1279   126   135*   65*
1289   144   150    74
1345    60    51*   23
1346    72    62*   27*
1347    84    75*   31*
1348    96    90    35*
1349   108   107*   39
1356    90    71*   33*
1357   105*   84*   38
1358   120    99    43
1359   135*  116*   48
1367   126    95    45*
1368   144   110    51
1369   162*  127*   57
1378   168   123    59
1379   189*  140*   66
1389   216   155    75
1456   120    78    34
1457   140*   91*   39
1458   160*  106*   44
1459   180   123    49*
1467   168   102    46
1468   192   117*   52
1469   216   134    58
1478   224*  130    60
1479   252   147    67*
1489   288   162*   76
1567   210   111    47
1568   240   126    53
1569   270   143    59
1578   280   139*   61*
1579   315*  156*   68
1589   360   171    77*
1678   336   150    62
1679   378   167*   69
1689   432   182    78
1789   504   195*   79*
2345   120    54*   26
2346   144    65*   30
2347   168    78    34
2348   192    93*   38
2349   216   110    42
2356   180    74*   36*
2357   210    87*   41*
2358   240   102    46
2359   270   119    51
2367   252    98*   48
2368   288   113*   54
2369   324*  130    60
2378   336   126    62
2379   378   143    69
2389   432   158    78
2456   240    81*   38
2457   280    94    43
2458   320*  109*   48
2459   360   126    53
2467   336   105    50
2468   384*  120*   56
2469   432   137*   62
2478   448*  133*   64
2479   504   150    71
2489   576   165    80
2567   420   114    52
2568   480   129*   58
2569   540   146*   64
2578   560*  142    66
2579   630   159*   73*
2589   720   174    82*
2678   672   153*   68
2679   756   170    75
2689   864   185*   84
2789  1008   198*   86
3456   360    86*   42
3457   420    99    47
3458   480   114    52
3459   540   131*   57
3467   504   110    54
3468   576   125*   60
3469   648*  142    66
3478   672   138*   68
3479   756   155    75
3489   864   170    84
3567   630   119    57
3568   720   134    63*
3569   810*  151*   69
3578   840   147    71
3579   945*  164*   78
3589  1080   179*   87
3678  1008   158    74
3679  1134*  175*   81*
3689  1296*  190*   90*
3789  1512   203*   93
4567   840   126    62
4568   960*  141*   68
4569  1080   158    74
4578  1120*  154*   76
4579  1260*  171    83*
4589  1440*  186*   92*
4678  1344*  165    80
4679  1512   182    87
4689  1728*  197*   96*
4789  2016*  210*  100*
5678  1680*  174    86
5679  1890*  191*   93
5689  2160*  206*  102*
5789  2520*  219*  107*
6789  3024*  230*  114*
 43 
1238    48*   78    26*
1249    72*  102    38*
1259    90*  111    47
1267    84*   90*   44*
1268    96*  105    50
1289   144   150    74
1358   120    99    43
1368   144   110    51
1378   168   123*   59
1389   216   155    75
1456   120    78    34
1467   168   102    46
1469   216   134*   58*
1567   210*  111    47
1568   240   126    53
1569   270   143    59
1678   336   150    62
1689   432   182    78
2347   168    78    34
2349   216   110    42*
2358   240   102    46
2359   270   119    51
2378   336   126    62
2379   378*  143    69*
2389   432   158    78
2457   280*   94*   43
2459   360*  126    53
2467   336   105    50
2479   504   150    71
2489   576*  165*   80*
2567   420   114    52
2679   756   170    75
3457   420    99    47
3458   480*  114    52
3467   504   110    54*
3479   756   155    75
3489   864*  170    84*
3567   630*  119    57*
3578   840   147*   71
3678  1008*  158    74
4567   840   126    62
4569  1080*  158    74
4679  1512*  182    87*
 22 
1289   144   150    74*
1358   120    99    43*
1368   144   110*   51
1389   216*  155    75
1456   120    78    34
1467   168   102    46
1568   240   126    53*
1569   270   143*   59*
1678   336   150    62
1689   432   182*   78
2347   168    78    34
2358   240   102    46
2359   270   119*   51
2378   336   126    62
2389   432   158*   78
2467   336   105*   50*
2479   504*  150    71*
2567   420   114*   52*
2679   756   170*   75
3457   420    99    47*
3479   756   155    75
4567   840*  126    62
 7 
1456   120*   78    34
1467   168   102    46
1678   336   150*   62
2347   168    78    34
2358   240*  102    46
2378   336   126*   62
3479   756*  155*   75*
 1 

  Posted by Charlie on 2004-12-26 21:02:25
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