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 A guess, can probably be improved.... by tomarken)

OK, so first we want to go as far as we can with just two stones.  We can do this with stones weighing 1, 2, 3, 9, and 16 pounds:

1
2
3
1+2
1+3
9-3
9-2
9-1
9
9+1
9+2
9+3
16-3
16-2
16-1
16
16+1
16+2
16+3

So we can get the values of 1 through 19 using just two of these stones.  This gives us a "range" of 39 more weights - in other words, if the next stone weighed x, we could weigh all the values from x-19 through x+19 just using the "x" stone and the combinations above. 

Before we get to that, however, we can extend the sequence first just using the first five stones:

16+3+1
16+3+2
16+9-3
16+9-2
16+9-1
16+9
16+9+1
16+9+2
16+9+3

So we can get all the way up to 28 with the first five stones.  If we choose the next stone to be 48, then we can cover the range from 29 up to 67 (48-19 up to 48+19).

Similarly, if we choose the seventh stone to be 87, then we can continue the range from 68 to 106 (87-19 to 87+19).

So the seven stones weigh 1, 2, 3, 9, 16, 48, and 87 pounds, and the maximum age is 106 years.

Edited on November 14, 2006, 3:18 pm

Posted by tomarken on 2006-11-14 15:04:42