bob and alan cross the desert (Posted on 2010-05-15)

	Submitted by broll
Rating: 3.0000 (1 votes)
Solution:	(Hide)
This is a variation of the old three jugs problem, which is supposed to have got Lagrange interested in mathematics. I noticed when reading around the subject that most of the problems deal with empty containers and a full (and sometimes inexhaustible) reservoir. If such problems are classed as 'emptyish', then the current problem is by contrast a 'fullish' problem, i.e. all 3 containers start with some contents. The first step in the solution is to realise that the problem is only soluble if the keg is full at the same time the other two containers hold one pint each. Since the size of the smallest container is known, we can also stipulate that the size of the second container must be relatively prime to the first, or no solution is possible ('it worked out in the end'.). This leaves 10,11,13,14,16, and 17 as possible sizes of the second container. Since there were 'a lot of steps', the sizes that rapidly produce a solution can be discarded. For example, the solution 10 simply requires the first container to be emptied into the keg, and the second into the first, leaving one pint in the second; when the first container is in turn emptied into the keg, one pint will be left in each container. Similarly 17 can be obtained by emptying the second container into the keg, emptying the first into the second, refilling the first and repeating, leaving one pint in the first container, after which the second can be emptied into the keg, leaving one pint in each container. The sizes 13 and 14 also yield quick solutions, as can readily be tested. This leaves Snark's solution of 16, and Dej Mar's solution of 11. Either theoretically yields a result in 21 steps; but as Dej Mar rightly points out, in order to remove all doubt, it is necessary first to ascertain the size of the larger container. If the larger container is of size 16, these steps are needed: (1)empty larger into keg, (2) empty smaller into larger; (3)fill smaller from keg; (4)empty smaller into larger; (5)when the larger is emptied into the keg, the keg is full,with 2 remaining in the larger, so we have determined that the larger keg is twice the size of the smaller,less 2 i.e. 16.The best we can now do is refill the larger container from the keg, thence leading to a result in 21 steps for a total of 27 steps. If the larger container is of size 11, these steps are needed: (1)empty smaller into keg: (2)fill smaller from larger; (3)empty smaller into keg, filling the keg, and the size of the larger container is now known to be 2 more than the smaller, in 3 steps. It now takes a further 19 steps to complete the solution, for a total of 21 steps. So the solution is that the larger container holds 16 pints.

	Subject	Author	Date
	Puzzle Thoughts	K Sengupta	2024-02-22 07:30:00
	re: Solution	broll	2010-07-01 11:07:49
	Solution	Dej Mar	2010-07-01 07:17:31
	re: Solution	broll	2010-06-30 08:07:32
	Solution	snark	2010-06-30 01:46:49
	re: solution?	broll	2010-05-21 11:37:23
	solution?	monica	2010-05-21 07:13:34