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?
The weights are powers of 3: 1, 3, 9, 27, 81, 243, 729. These can weight every integral weight up to 1+3+9+27+81+243+729=1093. The pattern can be seen in the weights up to 1+3+9+27=40. The solution has W as the weight to be determined, Left and Right show how to measure the weight. E.g. W=7 Left=W+3 Right=9+1 means put the grain plus the 3 in the left pan, and the 9 and the 1 in the right pan.
W=1 Left=W Right=1
W=2 Left=W+1 Right=3
W=3 Left=W Right=3
W=4 Left=W Right=3+1
W=5 Left=W+1+3 Right=9
W=6 Left=W+3 Right=9
W=7 Left=W+3 Right=9+1
W=8 Left=W+1 Right=9
W=9 Left=W Right=9
W=10 Left=W Right=9+1
W=11 Left=W+1 Right=9+3
W=12 Left=W Right=9+3
W=13 Left=W Right=9+3+1
W=14 Left=W+9+3+1 Right=27
W=15 Left=W+9+3 Right=27
W=16 Left=W+9+3 Right=27+1
W=17 Left=W+9+1 Right=27
W=18 Left=W+9 Right=27
W=19 Left=W+9 Right=27+1
W=20 Left=W+9+1 Right=27+3
W=21 Left=W+9 Right=27+3
W=22 Left=W+9 Right=27+3+1
W=23 Left=W+3+1 Right=27
W=24 Left=W+3 Right=27
W=25 Left=W+3 Right=27+1
W=26 Left=W+1 Right=27
W=27 Left=W Right=27
W=28 Left=W Right=27+1
W=29 Left=W+1 Right=27+3
W=30 Left=W Right=27+3
W=31 Left=W Right=27+3+1
W=32 Left=W+3+1 Right=27+9
W=33 Left=W+3 Right=27+9
W=34 Left=W+3 Right=27+9+1
W=35 Left=W+1 Right=27+9
W=36 Left=W Right=27+9
W=37 Left=W Right=27+9+1
W=38 Left=W+1 Right=27+9+3
W=39 Left=W Right=27+9+3
W=40 Left=W Right=27+9+3+1
W=41 Left=W+27+9+3+1 Right=81

Posted by Kevin
on 20070110 14:10:08 