A vessel with some head-space

For the "nut to crack" the concentration  1 / e (0.368) was found as the first result and e was found indirectly as the reciprocal value (dilution factor). With a head-space in vessel A it is possible to use another way of gradual approximation. The start concentration is unknown, but nevertheless it is possible to calculate a dilution factor. Instead of the sequence:

transfer - fill up - mix use add - mix - transfer

	In a variant with 4 steps, after addition of 1/4 of water to 4/4 of solution the dilution factor is 5/4 (1.25). After a repetition the dilution factor is 1.25 * 1.25. A total of 4 steps are possible until the equal-sized vessel B is full.
	Total dilution factor with vessel B full to the brim:
	With 4 steps:	1.254  =  (1 + 1/4)4	=  2.44
	With 10 steps:	1.110  =  (1 + 1/10)10	=  2.59
	With 100 steps:	(1 + 1/100)100	=  2.70
	With 1000 steps:	(1 + 1/1000)1000	=  2.72
	Up to infinite steps:	(1 + 1/n)n	=  2.718..
	The result is e and this formula is given in encyclopaedias, rather than the formula  1/e = (1 - 1/n)n, which was found in the "nut to crack".