Question 1162649
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Let "w" be the number of white widgets.


Then from the condition, you have this equation


    {{{w/(12+20+w)}}} = {{{1/p}}}.


It gives


    pw = 12 + 20 + w,   or

    (p-1)*w = 12 + 20 = 32.


So, "p-1" is one of the divisors of the integer number 32.


The possible divisors are p-1 = 1, 2, 4, 8, 16 and/or 32  (all degrees of 2, from 0 to 5).


So, there are 6 possible integer values for p,  2, 3, 5, 9, 17 and 33.    <U>ANSWER</U>
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Solved.