nasmuch as blossoms outnumber bees by a lot, the aptness of Mongut's metaphor resides in mathematics not entomology.  In the first place, worker bees are female.  Meanwhile, sophisticated solvers will doubtless answer the question in the puzzle as follows: 
 

If your answer is Case 1, you are in good -- at least plentiful -- company.  Case 1 was derived from a survey recently published by the U.S. Government, which concluded that men reported a median of seven female sex partners, while women reported a median of four male sex partners. 

According to an August 12, 2007 article in The New York Times, entitled "The Myth, the Math, the Sex" by Gina Kolata, in many research studies conducted all over the world, males report more sexual partners than females.

Conventional explanations based on natural selection will invariably embrace behavior differences consistent with Case 1.  However, in human populations wherein 'blossoms' and 'bees' are approximately equal in number, both Case 1 and Case 2 are just about mathematically impossible, as we shall see.

The subject of this puzzle is extremely popular among psychologists and important to researchers on health-care issues.  A web-search will confirm the availability of studies by the hundreds, each with lopsided results not much different from those on which the Materix puzzle is based.  Explanations vary widely.  Here begins our metanalysis...

Lopsided Population

We start by creating a matrix of a population, labeling the columns with male identities M1, M2, M3,...Mm and the rows with female identities F1, F2, F3,... Fn.  Let an entry of 1 in the matrix indicate a romantic relationship between the corresponding male and female -- a mating (hence the coinage here of the portmanteau 'materix').  After sprinkling in some 1s, here is how our population materix might look...
 

Now, we add up all the columns and divide by m then add up all the rows and divide by n, thereby finding the mating averages for m males and for n females.  If m =  n, of course, the averages come out the same for both genders -- whatever the pattern of entries.  That rules out both Case 1 and Case 2 and confirms our "both" as the solution to the Materix puzzle.  Or so it seems.

It is easy to show that for Case 1 to prevail, m /  n = 4 / 7. and that for Case 2, m /  n = 7 / 4.  For that amount of lopsidedness, there would have to be 75% more females than males in the population or vice versa.  Give the previous sentence exclamatory punctuation if you like. 

Lopsided Sample

Both the Materix puzzle and the survey by the U.S. Government postulated... 

(a) "questionnaires completed by an equal number of males and females" and...
(b) responders "selected at random from the same population." 

By the way, you may have noticed that none of the males and females in that particular sampling  materix ever mated with each other.   Whenever that takes place, the entry of 1 applies to both genders equally, shifting the averages toward each other. 

rothels are often cited as exceptions, since they trade in transitory romantic relationships for males while not bumping up the averages for females within the population. Please note, however, the magnitude of the seven-to-four ratio requires the existence of a huge romance-for-finance industry, at the same time excluding prostitutes from the population materix.  Oh, and what about female passengers on cruise ships and gigolos at ports o' call? 

Nevertheless, several reseachers offer prostitution as an explanation for the lopsidedness.  Entries outside the sampling materix, whether occurring in bordellos or motel rooms, will indeed have the power to skew the averages one way or the other. Let us suppose for the moment that the sampling materix does accurately represent the population materix.  Some people will insist that, if m = n, the lopsidedness in Case 1 means that a large number of males must simply be selecting romantic partners from outside the population materix.  Sure, and females don't ever do that? 

Here's the kicker: statistics for partners outside of the population materix really ought to be tabulated along with their own romantic experiences, which effectively appropriates their outside partners into the population materix.  In concept, then, both m and n may be driven to increase exponentially, limited ultimately by the carrying capacity of the planet whereupon m = n and neither Case 1 nor Case 2 can prevail.

Lopsided Desires

It may be instructive to note that one study did not inquire about cumulative life-long mating experiences but about desires instead.  The question was of the form, "How many romantic relationships do you want to have over the next year?"  Statistical differences in the gender-specific responses were quite pronounced: males saying "many" and females saying "only one."

Lopsided Orientation

Each romantic relationship being considered on this page assumes the participation of one gynotaxic male and one androtaxic female.   Suppose that there is a common flaw herein and also in the worldwide studies -- that they all inadvertently included same-sex partnerships.  Such romantic relationships would count twice for androtaxic males and zero for females.  Or vice versa -- twice for gynotaxic females and zero for males.  The explanation for the lopsidedness in Case 1 would then depend on the relative frequencies of each sexual orientation in the population compounded with the lopsided distribution of romantic partners reported by males and females in each category. 

Sophisticated solvers endeavor not to offer explanations for statistical anomalies by introducing statistical unknowables.

Lopsided Eligibility

What about age?  Curiously, the research to date appears to neglect this question. Unmistakable gender asymmetry is commonly observed, with males at any age favoring romantic relationships with females at younger ages while females at any age being potentially attracted to males of every age.  Let us visualize that the questionnaires are segregated into stacks according to age. 

The study would surely find that young men have fewer opportunities for romantic relationships than young women.  A typical male’s mating prospects will start out limited mainly to females at his own age and then increase with age. 

A young female, on the other hand, has more abundant choices than a young male.  Beginning at the earliest adult age, a female has romantic access to the whole population of males, although young chicks will seldom go for old codgers when there are plenty of young studs around. 

ost of the statistical studies you will find on the web take place in university settings.  An eligibility materix limited in scope to young people will surely have m > n, which favors Case 2 not Case 1.  As males get older, the number of eligible females gets steadily larger and so, presumably, will the reported quantity of romances with them.  Meanwhile, unlike the male, the female enjoys a target population of eligible males that will decline steadily with her own age. 

An eligibility materix of middle-aged people will probably have m < n, which does favor Case 1 -- but only if it's not too late.  Matrimony and parenting will put the statistical brakes on accumulating romantic relationships by both males and females. 

Lopsided Median 

Sophisticated solvers may notice that the Materix puzzle statement used the word "typical" ("typical males"..."typical females").  You are invited to interpret that phraseology as merely an informal way of saying "average" or mean.  Oh, but then it is surely fair to take "typical" to mean median, as explicitly depicted in the U.S. Government report.

Some writers have tried to explain gender asymmetry in terms of the mathematical distinctions that can arise between median and mean. That subject is treated at length in Revenge of the DAR, where the power of the median is seen to discard outliers (trollops and Lotharios) in estimating "central tendency" -- exactly the opposite of what one needs for explaining the lopsidedness in either Case 1 or Case 2. 

Lopsided Reporting

One final parsing of the Materix puzzle statement brings the word "reported" into consideration.  And for good reason.  The data in all of the studies necessarily comes from the subjects themselves, with no feasible corroboration. 

Several writers argue that gender-specific standards of deportment for many cultures will account for the differences in the reported numbers -- that if asked, a man, believing he will be admired for having accumulated a lot of romantic "conquests," may have an incentive to exaggerate, and a woman, expecting to garner more approval for having withheld her charms, will tend to minimize her numbers.

Some studies blame differences on the value assigned by the respective participants to past romantic relationships.  Women are said to remember the specifics of each and every relationship, while men remember only generalities, which allows plenty of fuzziness in estimating the actual number. 

Lopsided Anonymity

One particular study may have teased some statistical truth out of the reported numbers using anonymity as the independent variable.  The sample-sizes (in the dozens) were too small to be any more than suggestive; nevertheless, here is the protocol. 

Half of the males and half of the females were assured that their questionnaires would be kept anonymous.  The others were told that the identities of participants would be disclosed at the end of the study.  The results were striking.  Almost deserving an exclamation point.

• In the anonymous group, the quantities of reported romantic relationships were about the same for both males and females -- between six and seven different partners for each volunteer. 
• On the non-anonymous questionnaires, males reported about the same quantity of romantic partners -- if anything, higher than the anonymous group, within the accuracy afforded by the small sample-size. 
• Meanwhile, the females who were deprived of anonymity, reported about half the quantities of romantic relationships. 

hus do we try to explain objective impossibilities by conjecturing about subjective possibilities.  In the words of King Mongut of Siam, "There are times I almost think I am not sure of what I absolutely know."