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Dear Mitch,

I was wondering if you would post one of those balance-scale problems like you've done in the past for a lot of the holidays, only this one for St. Patrick's Day?

Thank you,

Ms. B. Laurent


Dear Ms. B. Laurent,


It would be my pleasure to post such a problem.  In fact, those particular ones, the 'balance-scale problems', have turned out to be far more popular than I ever could have anticipated!


A series of weighings is shown here. The items on the scales are toys. There are three different kinds of toys – 'Framed pictures' of leprechauns, plastic cakes iced with the words "Happy St. Valentine's day," and three-leaf clovers, known as the "Shamrock".


Going from top to bottom, the first two scales (scale A and scale B) are balanced and complete.


Scale A depicts five toy cakes on its left-hand side. And it depicts one Shamrock and one picture of a leprechaun on its right-hand side.


Scale B depicts one Shamrock on its left-hand side. And it depicts one picture of a leprechaun and one cake on its right-hand side.


The third scale, however, (Scale C), is not yet balanced.


QUESTION: How many 'framed pictures' of the leprechaun will have to be placed on the right-hand side of Scale C in order to form a balance with the left-hand side of that scale, which, as depicted in the illustration, is holding 1 toy cake and 7 Shamrocks? 


Mathematically yours,