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King's Maths School

Challenge 451: Efficient Weights

How many weights can a weigher weigh with just four weights? Thanks to KCLMS student Nathan for this week's challenge!

(Apologies for the missing challenge last week - things got busy in WMCville!)

(1)

I have four weights of integer value and a set of balance scales.

(These scales have two pans; I can put an object into each pan and see which is heavier, or if they are the same weight. Google it if you don't know what I'm talking about!)

An object is put into one of the pans, and I am allowed to put any of my weights into the other pan. By choosing different combinations of my weights, I can determine the weight of any object with integer weight up to N.

Given that N is the largest possible value for which this statement can be true, what are the values of my weights?

(2)

The situation is the same, but with one difference: I can put my weights into either pan of the scales. 

N is once again the largest possible value such that I can determine any integer weight up to N. What are the values of my weights?

Try to give a convincing justification of your answers!

 

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