A man has a physical balance and a stone of mass 40 kg . He divides the stone into four parts A,B,C,D of mass a kg , b kg , c kg , d kg respectively , such that using A,B,C,D he can measure any specific integral mass from 1 kg to 40 kg . Find the mass of all four parts .i.e., a kg , b kg , c kg , d kg ? {The answer to this question is 1 kg , 3 kg , 9 kg , 27 kg .Explanation: if that man is asked to measure 2kg he will keep 3kg on one of the palm of physical balance and 1 kg on the other . So the net weight being (3-1) kg = 2 kg. For measuring 4 kg he will keep he will use 3kg and 1kg . For measuring 5 kg he will keep 9 kg on one side and 4 kg ( 3kg+1kg ) on the other side . In the same way , he can measure any of the following weights : 1 kg , 2kg , 3kg , ............ 39 kg , 40 kg using those four parts . }. I found this question in a magazine .Though I was able to solve this question using heat and trial method , I did not understand how to solve this question mathematically using algebra or some other convenient mathematical method. Can anyone please solve his question mathematically
							
A man has a physical balance and a stone of mass 40 kg . He divides the stone into four  parts A,B,C,D of mass a kg , b kg , c kg , d kg respectively , such that using A,B,C,D he can measure any specific integral  mass from 1 kg to 40 kg . Find the mass of all four parts .i.e., a kg , b kg , c kg , d kg ? {The answer to this question is 1 kg , 3 kg , 9 kg , 27 kg .Explanation: if that man is asked to measure 2kg he will keep 3kg on one of the palm of physical balance and 1 kg on the other . So the net weight being (3-1) kg = 2 kg. For measuring 4 kg he will keep he will use 3kg and 1kg . For measuring 5 kg he will keep 9 kg on one side and 4 kg ( 3kg+1kg ) on the other side . In the same way , he can measure any of the following weights : 1 kg , 2kg , 3kg , ............ 39 kg , 40 kg using those four parts .  }.   I found this question in a magazine .Though I was able to solve this question using heat and trial method , I did not understand how to solve this question mathematically using algebra or some other convenient mathematical method. Can anyone please solve his question mathematically