perplexus.info :: Weights and Scales : Ceremonial Stones

Ceremonial Stones (Posted on 2006-11-14)

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 trang	2019-01-03 03:17:19
re(2): Rebuttal	FrankM	2008-02-09 01:17:17
re: Rebuttal	Bractals	2008-02-08 14:09:11
Rebuttal	FrankM	2008-02-08 10:06:37
re: That's peculiar	Bractals	2008-02-03 12:45:22
That's peculiar	FrankM	2008-02-03 11:55:53
Generalisation	FrankM	2008-02-03 11:46:36
Disagree with "Disagree With Solution - Max Age Very Large"	jaime	2007-01-30 17:39:54
re: Keeping the Tradition	Gamer	2007-01-10 15:07:00
Disagree With Solution - Max Age Very Large	Kevin	2007-01-10 14:10:08
re(2): Max age - A computer exploration	Bernie Hunt	2006-11-20 09:36:21
re: Max age - A computer exploration	Dej Mar	2006-11-20 02:24:30
Max age: 115	Bernie Hunt	2006-11-15 10:20:16
re(2): A computer exploration -- solution???	Charlie	2006-11-15 08:56:13
re: A computer exploration -- solution???	Bractals	2006-11-15 03:15:02
re(3): upper limit	Bractals	2006-11-15 03:07:33
re: A computer exploration -- solution??? -- how to get the numbers	Charlie	2006-11-14 23:50:02
A computer exploration -- solution???	Charlie	2006-11-14 23:42:00
re(2): upper limit	Charlie	2006-11-14 23:33:40
re: upper limit	Bractals	2006-11-14 22:13:33
re(2): At least 109 years	Bernie Hunt	2006-11-14 18:17:58
re: At least 109 years	Bractals	2006-11-14 18:07:48
At least 109 years	Bernie Hunt	2006-11-14 17:06:03
An improved guess...	tomarken	2006-11-14 15:04:42
A guess, can probably be improved....	tomarken	2006-11-14 13:55:29
upper limit	Charlie	2006-11-14 13:45:30

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