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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 computer solution | Comment 1 of 4

There are 126 numbers that meet the original criteria:

 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

Shown are the number, the product of the digits, the sum of the squares of the digits, and the sum of the pairwise products of the first two and last two.

Of the above 43 numbers have all three derived quantities that are non-unique:

 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

(They may be unique in one or two columns in the listing immediately above, as the match might be with a number that has some other column with a unique value.)

Then the second round takes place and we are down to only those that have all three non-unique values in this list.  There are 22 such numbers:

 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

Then, repeating for the third round, there are 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

We know that after this, all three announced the answer, so it must be 3479, which is the only one with all unique values among those remaining.

DEFDBL A-Z
DIM n(126), p(126), ssq(126), s2(126)
OPEN "inherit.txt" FOR OUTPUT AS #2
FOR d1 = 1 TO 6
FOR d2 = d1 + 1 TO 7
FOR d3 = d2 + 1 TO 8
FOR d4 = d3 + 1 TO 9
  i = i + 1
  n(i) = d1 * 1000 + d2 * 100 + d3 * 10 + d4
  p(i) = d1 * d2 * d3 * d4
  ssq(i) = d1 * d1 + d2 * d2 + d3 * d3 + d4 * d4
  s2(i) = d1 * d2 + d3 * d4
  ct = ct + 1
NEXT
NEXT
NEXT
NEXT
PRINT #2, ct

FOR i = 1 TO ct
 PRINT #2, n(i), p(i), ssq(i), s2(i)
NEXT
PRINT #2,

REDIM multi(3, ct)
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF p(i) = p(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(1, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF ssq(i) = ssq(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(2, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF s2(i) = s2(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(3, i) = 1
NEXT

newCt = 0
FOR i = 1 TO ct
  IF multi(1, i) AND multi(2, i) AND multi(3, i) THEN
    newCt = newCt + 1
    n(newCt) = n(i): p(newCt) = p(i): ssq(newCt) = ssq(i): s2(newCt) = s2(i)
  END IF
NEXT
ct = newCt
PRINT #2, ct

FOR i = 1 TO ct
 PRINT #2, n(i), p(i), ssq(i), s2(i)
NEXT
PRINT #2,


REDIM multi(3, ct)
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF p(i) = p(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(1, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF ssq(i) = ssq(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(2, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF s2(i) = s2(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(3, i) = 1
NEXT

newCt = 0
FOR i = 1 TO ct
  IF multi(1, i) AND multi(2, i) AND multi(3, i) THEN
    newCt = newCt + 1
    n(newCt) = n(i): p(newCt) = p(i): ssq(newCt) = ssq(i): s2(newCt) = s2(i)
  END IF
NEXT
ct = newCt
PRINT #2, ct

FOR i = 1 TO ct
 PRINT #2, n(i), p(i), ssq(i), s2(i)
NEXT
PRINT #2,

REDIM multi(3, ct)
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF p(i) = p(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(1, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF ssq(i) = ssq(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(2, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF s2(i) = s2(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(3, i) = 1
NEXT

newCt = 0
FOR i = 1 TO ct
  IF multi(1, i) AND multi(2, i) AND multi(3, i) THEN
    newCt = newCt + 1
    n(newCt) = n(i): p(newCt) = p(i): ssq(newCt) = ssq(i): s2(newCt) = s2(i)
  END IF
NEXT
ct = newCt
PRINT #2, ct

FOR i = 1 TO ct
 PRINT #2, n(i), p(i), ssq(i), s2(i)
NEXT

REDIM multi(3, ct)
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF p(i) = p(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(1, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF ssq(i) = ssq(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(2, i) = 1
NEXT
FOR i = 1 TO ct
  t = 0
  FOR j = 1 TO ct
    IF s2(i) = s2(j) THEN t = t + 1
  NEXT
  IF t > 1 THEN multi(3, i) = 1
NEXT

newCt = 0
FOR i = 1 TO ct
  IF (multi(1, i) OR multi(2, i) OR multi(3, i)) = 0 THEN
    newCt = newCt + 1
    n(newCt) = n(i): p(newCt) = p(i): ssq(newCt) = ssq(i): s2(newCt) = s2(i)
  END IF
NEXT
ct = newCt
PRINT #2, ct

FOR i = 1 TO ct
 PRINT #2, n(i), p(i), ssq(i), s2(i)
NEXT
PRINT #2,


  Posted by Charlie on 2004-12-25 01:18:08
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