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Ceremonial Stones (Posted on 2006-11-14) Difficulty: 5 of 5
A certain tribe had an interesting kind of annual festival, in which every male member of the tribe (regardless of age) had to contribute a levy of grain into the tribal store. Their unit of weight was roughly the same as our pound avoirdupois, and each tribesman had to contribute one pound of grain for every year of his age.

The contributions were weighed on the tribe's ceremonial scales, using a set of seven ceremonial stones. Each of these weighed an integral number of pounds, and it was an essential part of the ritual that not more than three of them should be used for each weighing, though they need not all be in the same pan.

If ever a tribesman lived to such an age that his contribution could no longer be weighed by using three or fewer stones, the levy of grain would terminate forever. And in the previous year, one old man had died only a few months short of attaining this critical age, greatly to the relief of the headman of the tribe.

It has been determined that the stones can measure the maximum age. What is this age and what were the weights of the seven ceremonial stones?

  Submitted by Bractals    
Rating: 3.0000 (7 votes)
Solution: (Hide)
The maximum age is 122 and the weights are

1, 3, 7, 12, 43, 76, 102

For a discussion of this problem see pages 5-7 of Optima 65 here.

Below are weighings for ages 1-122 where a minus sign means the weight is in the pan with the grain and the plus sign means the weight is in the other pan:
   
001 = +001        
002 = +003-001     = +012-007-003
003 = +003         = +007-003-001
004 = +003+001     = +007-003     = +012-007-001
005 = +007-003+001 = +012-007    
006 = +007-001     = +012-007+001
007 = +007        
008 = +007+001     = +012-003-001 = +012-007+003
009 = +007+003-001 = +012-003    
010 = +007+003     = +012-003+001
011 = +007+003+001 = +012-001    
012 = +012        
013 = +012+001    
014 = +012+003-001 = +102-076-012
015 = +012+003    
016 = +012+003+001 = +012+007-003
017 = -102+076+043
018 = +012+007-001
019 = +012+007     = +102-076-007
020 = +012+007+001
021 = +076-043-012
022 = +012+007+003
023 = +102-076-003
024 = +043-012-007
025 = +102-076-001
026 = +076-043-007 = +102-076    
027 = +102-076+001
028 = +043-012-003
029 = +102-076+003
030 = +043-012-001 = +076-043-003
031 = +043-012    
032 = +043-012+001 = +076-043-001
033 = +043-007-003 = +076-043     = +102-076+007
034 = +043-012+003 = +076-043+001
035 = +043-007-001
036 = +043-007     = +076-043+003
037 = +043-007+001
038 = +043-012+007 = +102-076+012
039 = +043-003-001 = +043-007+003
040 = +043-003     = +076-043+007
041 = +043-003+001
042 = +043-001    
043 = +043        
044 = +043+001    
045 = +043+003-001 = +076-043+012
046 = +043+003    
047 = +043+003+001 = +043+007-003 = +102-043-012
048 = +043+012-007
049 = +043+007-001
050 = +043+007    
051 = +043+007+001
052 = +043+012-003 = +102-043-007
053 = +043+007+003
054 = +043+012-001
055 = +043+012    
056 = +043+012+001 = +102-043-003
057 = +076-012-007
058 = +043+012+003 = +102-043-001
059 = +102-043    
060 = +102-043+001
061 = +076-012-003
062 = +043+012+007 = +102-043+003
063 = +076-012-001
064 = +076-012    
065 = +076-012+001
066 = +076-007-003 = +102-043+007
067 = +076-012+003
068 = +076-007-001
069 = +076-007     = +102-076+043
070 = +076-007+001
071 = +076-012+007 = +102-043+012
072 = +076-003-001 = +076-007+003
073 = +076-003    
074 = +076-003+001
075 = +076-001    
076 = +076        
077 = +076+001    
078 = +076+003-001
079 = +076+003    
080 = +076+003+001 = +076+007-003
081 = +076+012-007
082 = +076+007-001
083 = +076+007     = +102-012-007
084 = +076+007+001
085 = +076+012-003
086 = +076+007+003
087 = +076+012-001 = +102-012-003
088 = +076+012    
089 = +076+012+001 = +102-012-001
090 = +102-012    
091 = +076+012+003 = +102-012+001
092 = +102-007-003
093 = +102-012+003
094 = +102-007-001
095 = +076+012+007 = +102-007    
096 = +102-007+001
097 = +102-012+007
098 = +102-003-001 = +102-007+003
099 = +102-003    
100 = +102-003+001
101 = +102-001    
102 = +102        
103 = +102+001    
104 = +102+003-001
105 = +102+003    
106 = +102+003+001 = +102+007-003
107 = +076+043-012 = +102+012-007
108 = +102+007-001
109 = +102+007    
110 = +102+007+001
111 = +102+012-003
112 = +076+043-007 = +102+007+003
113 = +102+012-001
114 = +102+012    
115 = +102+012+001
116 = +076+043-003
117 = +102+012+003
118 = +076+043-001
119 = +076+043    
120 = +076+043+001
121 = +102+012+007
122 = +076+043+003
Given the seven weights it's easy to generate this list. Finding the seven weights is the problem.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(3): A computer exploration -- solution???thu trang2019-01-03 03:17:19
Questionre(2): RebuttalFrankM2008-02-09 01:17:17
re: RebuttalBractals2008-02-08 14:09:11
Some ThoughtsRebuttalFrankM2008-02-08 10:06:37
re: That's peculiarBractals2008-02-03 12:45:22
That's peculiarFrankM2008-02-03 11:55:53
GeneralisationFrankM2008-02-03 11:46:36
Disagree with "Disagree With Solution - Max Age Very Large"jaime2007-01-30 17:39:54
re: Keeping the TraditionGamer2007-01-10 15:07:00
Disagree With Solution - Max Age Very LargeKevin2007-01-10 14:10:08
re(2): Max age - A computer explorationBernie Hunt2006-11-20 09:36:21
Solutionre: Max age - A computer explorationDej Mar2006-11-20 02:24:30
Max age: 115Bernie Hunt2006-11-15 10:20:16
re(2): A computer exploration -- solution???Charlie2006-11-15 08:56:13
re: A computer exploration -- solution???Bractals2006-11-15 03:15:02
re(3): upper limitBractals2006-11-15 03:07:33
Hints/Tipsre: A computer exploration -- solution??? -- how to get the numbersCharlie2006-11-14 23:50:02
Hints/TipsA computer exploration -- solution???Charlie2006-11-14 23:42:00
re(2): upper limitCharlie2006-11-14 23:33:40
Questionre: upper limitBractals2006-11-14 22:13:33
re(2): At least 109 yearsBernie Hunt2006-11-14 18:17:58
Some Thoughtsre: At least 109 yearsBractals2006-11-14 18:07:48
At least 109 yearsBernie Hunt2006-11-14 17:06:03
SolutionAn improved guess...tomarken2006-11-14 15:04:42
SolutionA guess, can probably be improved....tomarken2006-11-14 13:55:29
Some Thoughtsupper limitCharlie2006-11-14 13:45:30
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