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 Inheritance Envelopes (Posted on 2004-12-24)
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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 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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