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?

(In reply to Max age: 115 by Bernie Hunt)

Using Visual Basic, I was able to find an age of 119 with seven ceremonial stones of 1, 2, 4, 24, 31, 72 and 86 pounds.  Therefore, the age at which would terminate the grain may be 120 years.

s1 = 1     s2 = 2     s3 = 4     s4 = 24
s5 = 31   s6 = 72   s7 = 86     

In Scale 1 is the levy of grain in pounds and zero, one or two stones to be balanced against the one, two or three stones in Scale 2.  Note, only three stones total are used in both scales. 

Scale 1          Scale 2
  1            = s1
  2            = s2
  3            = s1 + s2
  4            = s3
  5            = s1 + s3
  6            = s2 + s3
  7 + s4       = s5
  8 + s4       = s1 + s5
  9 + s4       = s2 + s5
 10 + s3 + s6  = s7
 11 + s4       = s3 + s5
 12 + s2 + s6  = s7
 13 + s1 + s6  = s7
 14 + s6       = s7
 15 + s6       = s1 + s7
 16 + s6       = s2 + s7
 17 + s4 + s5  = s6
 18 + s6       = s3 + s7
 19 + s1 + s3  = s4
 20 + s3       = s4
 21 + s3       = s1 + s4
 22 + s2       = s4
 23 + s1       = s4
 24            = s4
 25            = s1 + s4
 26            = s2 + s4
 27 + s3       = s5
 28            = s3 + s4
 29 + s2       = s5
 30 + s1       = s5
 31            = s5
 32            = s1 + s5
 33            = s2 + s5
 34            = s1 + s2 + s5
 35            = s3 + s5
 36            = s1 + s3 + s5
 37            = s2 + s3 + s5
 38 + s6       = s4 + s7
 39 + s2 + s5  = s6
 40 + s1 + s5  = s6
 41 + s5       = s6
 42 + s5       = s1 + s6
 43 + s5       = s2 + s6
 44 + s3 + s4  = s6
 45 + s6       = s5 + s7
 46 + s2 + s4  = s6
 47 + s1 + s4  = s6
 48 + s4       = s6
 49 + s4       = s1 + s6
 50 + s4       = s2 + s6
 51 + s3       = s4 + s5
 52 + s4       = s3 + s6
 53 + s2       = s4 + s5
 54 + s1 + s5  = s7
 55            = s4 + s5
 56            = s1 + s4 + s5
 56 + s5       = s1 + s7
 57            = s2 + s4 + s5
 58 + s3 + s4  = s7
 59            = s3 + s4 + s5
 60 + s2 + s4  = s7
 61 + s1 + s4  = s7
 62 + s4       = s7
 63 + s4       = s1 + s7
 64 + s4       = s2 + s7
 65 + s5       = s4 + s6
 66 + s4       = s3 + s7
 67 + s1 + s3  = s6
 68 + s3       = s6
 69 + s3       = s1 + s6
 70 + s2       = s6
 71 + s1       = s6
 72            = s6
 73            = s1 + s6
 74            = s2 + s6
 75            = s1 + s2 + s6
 76            = s3 + s6
 77            = s1 + s3 + s6
 78            = s2 + s3 + s6
 79 + s5       = s4 + s7
 80 + s2 + s3  = s7
 81 + s1 + s3  = s7
 82 + s3       = s7
 83 + s3       = s1 + s7
 84 + s2       = s7
 85 + s1       = s7
 86            = s7
 87            = s1 + s7
 88            = s2 + s7
 89            = s1 + s2 + s7
 90            = s3 + s7
 91            = s1 + s3 + s7
 92            = s2 + s3 + s7
 93 + s4       = s5 + s7
 94 + s2       = s4 + s6
 95 + s1       = s4 + s6
 96            = s4 + s6
 97            = s1 + s4 + s6
 98            = s2 + s4 + s6
 99 + s3       = s5 + s6
100            = s3 + s4 + s6
101 + s2       = s5 + s6
102 + s1       = s5 + s6
103            = s5 + s6
104            = s1 + s5 + s6
105            = s2 + s5 + s6
106 + s3       = s4 + s7
107            = s3 + s5 + s6
108 + s2       = s4 + s7
109 + s1       = s4 + s7
110            = s4 + s7
111            = s1 + s4 + s7
112            = s2 + s4 + s7
113 + s3       = s5 + s7
114            = s3 + s4 + s7
115 + s2       = s5 + s7
116 + s1       = s5 + s7
117            = s5 + s7
118            = s1 + s5 + s7
119            = s2 + s5 + s7

Edited on November 21, 2006, 1:42 pm

Posted by Dej Mar on 2006-11-20 02:24:30