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Dear Mitch,

I like the balance-scale math puzzles you post at holiday time, and I just finished solving the newest one you put up for Halloween.  This year, for the first time in ten years, instead of teaching high school math, I'm teaching fifth graders at a new elementary school.  So would it be possible for you to post an easier math puzzle in time for this Halloween so I can introduce it to my students without having to worry that they'll become too frustrated to solve it?

(And the best case scenario would be if it could be one of your balance-scale problems that I see on some of the Japanese sites, but you also do.)

Thank you,

Mrs. J 


Dear Mrs. J.,

Of course!

The diagram of the three scales for this season's Halloween puzzle for your class of fifth graders (and also suitable for mathematicians of all ages to try. . .) can be seen below (just scroll down until you see it!).

Here is the worded part that you will need to read along with the diagram:

A series of weighings is shown here.  On the scales are two kinds of cookies made for a Halloween party.  One cookie is in the shape of a bat.  The other cookie is in the shape of a pumpkin.  Going from top to bottom, the first two scales (scale A and scale B) are balanced and complete, but the third scale (scale C) is not.  

QUESTION:  Based on what you can learn from looking at the way the first two scales are balanced, how many bat cookies must be placed on the empty side of scale C in order to form a balance with the left-hand side of that scale (Which, as shown in the picture, is weighed down by 5 bat cookies and 1 pumpkin cookie)?

Both kinds of cookies are made with a soft line down the back so that each cookie can easily be broken into two even halves.  So. . . you can use half-cookies, such as half a bat-shaped cookie, to come up with your answer).  GOOD LUCK!!