Our July Insights puzzle explored one of biology’s greatest mysteries: Why do most large, complex animals come in two sexes? We received some excellent responses, including a formal mathematical proof for Problem 1 by Austin Joey Anderson, ready to be submitted to a professional journal!

As strange as it might seem, answering this biology question with a formal proof makes good sense. Underlying the biology is a math problem extending the concept of parity, or the property of an integer that assigns it to one of two categories: even or odd. Here we use an imaginary species of chameleon to consider how something analogous to parity works when there are three categories.

Our first problem concerns an interesting mathematical conundrum that could arise in the case of three sexes. Suppose a new species of chameleon was discovered in the Southwestern United States on July 4, the first species found to have three sexes. This species has been aptly named Chamaeleo americanus patrioticus for reasons we specify below.

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The adults of this Chamaeleo americanus patrioticus are colored fully red, white or blue, and this defines their sex at a given time. Being chameleons, however, they can and do change color, and therefore their sex. Specifically, when two differently colored individuals meet face to face, as happens quite frequently, they both change to the third color, as you can see in the illustration shown here. The chameleons retain this new color till the next such encounter with an individual of a different color. In effect, the color change is like blushing — a signal that says “not now.” There could be hundreds of such blushing encounters before the chameleons of two different colors are familiar enough to mate.  What concerns us is that, for an animal breeder, this behavior can result in the dreaded problem of unicolorization before breeding can even start. If these color-changing encounters cause all your chameleons to change to the same color, you are out of luck — same-colored individuals cannot reproduce with each other. Whether unicolorization can happen or not in a given group of chameleons is crucially dependent on the number of individuals of each color that you start out with. Some sets of numbers almost invariably result in unicolorization of the group sooner or later, while others never do. Can you predict whether unicolorization can happen if you start with the numbers of these chameleons specified in A, B and C below, and if so, how?

A) 8 red, 5 white and 14 blue;
B) 9 red, 10 white and 16 blue;
And finally, of course:
C) 7 red, 6 white and 50 blue

Based on the above, can you figure out a general rule on how many individuals of each of the three colors you should start out with, so that the group is resistant to unicolorization?

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A monthly puzzle celebrating the sudden insights and unexpected twists of scientific problem solving. Your guide is Pradeep Mutalik, a medical research scientist at the Yale Center for Medical Informatics and a lifelong puzzle enthusiast.

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There is indeed an elegant rule for the property that makes a group resistant to unicolorization: the number of individuals of each color must have a different remainder when divided by 3. In mathematical terms, the three numbers must yield different values modulo 3, thus making a complete set of such values: 0 modulo 3, 1 modulo 3 and 2 modulo 3. We can think of these as natural extensions to the concept of parity when there are three distinct types of numbers: instead of odd and even numbers, we now have to think of parity 0, parity 1 and parity 2 numbers. As shown in proofs submitted by Sayantan Khan, Mark Pearson and Austin Joey Anderson, not only is it sufficient to have all three types of numbers to avoid any possibility of unicolorization, but it is also necessary. In other words, if the three numbers don’t have this property, you cannot avoid unicolorization under all possible circumstances. I hope you find the following informal proofs intuitive.

First, let’s consider this condition in preventing unicolorization. Assume that the group starts with numbers that constitute a complete set of parities modulo 3 (i.e., have remainders 0, 1 and 2 when divided by 3). Then any color-changing encounter causes two of the numbers to decrease by 1 and the third one to increase by 2. But in modulo 3 arithmetic, an increase of 2 is the same as a decrease of 1! So all three numbers decrease by 1 modulo 3, meaning all three numbers still belong to the three different categories modulo 3. So no matter how many color-changing encounters take place, the three numbers continue to form a complete set modulo 3. For unicolorization to take place, you must make two colors disappear; both have to be 0 modulo 3. But as we showed above, that is impossible. So any combination of numbers that belong to all three categories modulo 3 cannot unicolorize under any circumstances.

To prove that such a combination of numbers is necessary, we have to show that any triplet that does not have the above property cannot avoid unicolorization under all circumstances. If the numbers do not form a complete set modulo 3, then at least two of them, let’s say a and b, must have the same value modulo 3. They are either equal or they differ by a multiple of 3. Let a be less than or equal to b (a ≤ b), and let us call the third number c. The italicized values a, b and c reflect the numbers of chameleons belonging to the colors a, b and c, whatever they may be. Now colors a and b should have repeated encounters till the number of a’s becomes 0. If a and b are equal, then unicolorization is complete. If not, the number of b’s is now a multiple of 3, and type c has increased from its original value by 2a. With the numbers of the three colors now at the new values A = 0, B and C, we execute the following cycle of three encounters: b:c, a:b, a:b. Note that the b:c encounter produces two new individuals of color a, who can take part in the second and third encounters. After the first step, we have the three types at 2, B – 1, C – 1. After the second step, they are at 1, B – 2, C + 1. After the final step, color a is back to 0, and the other two are at B – 3 and C + 3, respectively. So the net effect of the three-step cycle is to decrease the number of b’s by 3 and increase the c’s by 3. Like a virtual particle in physics, the type a’s arise from nothingness, do their work and then disappear back to nothingness. Continue repeating this three-step cycle until b reaches 0. Voilà, unicolorization is complete!

Let’s apply this method to the sets of numbers given (noting that many methods can achieve unicolorization):

A) 8, 5 and 14 are all 2 mod 3. Put a=5, b=8 and c=14. After five a:b encounters, we have 0, 3 and 24. After one cycle of b:c, a:b, a:b, we have unicolorization with the numbers changing after each step, as follows: 2, 2, 23; 1, 1, 25; and 0, 0, 27.

B) Here, 10 and 16 are both 1 mod 3, while 9 is 0 mod 3. So a=10, b=16 and c=9. After 10 a:b encounters, we have 0, 6 and 29. Two cycles of b:c, a:b, a:b give unicolorization to 0, 0 and 35.

C) This combination comes from the American flag: seven red stripes, six white stripes and 50 “blue” stars. This is a complete set modulo 3: 7 is 1 mod 3; 6 is 0 mod 3; and 50 is 2 mod 3, and hence this combination cannot be unicolorized.

While this problem is a mathematical contrivance that has nothing to do with real biology, our second and third problems explore a possible real reason why asexual species fail, despite their reproductive advantages. Asexual species can fall prey to something that I called “the fatal flaw of serial multiplication” — a principle that was illustrated in Problem 2. If you haven’t tried this problem, I encourage you to do so — it includes hints and the answer itself, in case you need them.

In Problem 3, we applied this principle to comparing the population growth of two species, one sexual and one asexual.

Imagine two species of lizards, one reproducing sexually and the other asexually. Imagine that they have reached somewhat stable populations to the limits of their available resources, so that their growth rates from generation to generation are only slightly above 1. The asexual species will still have a higher mean population growth rate, but this growth rate will also be more variable. Let’s say the mean population growth rate from one generation to the next for the sexual species is 1.1 with a standard deviation of 0.15, and the rate for the asexual species is 1.2 with a standard deviation of 0.3 (replace negative growth rates by zero). The population of the asexual species grows far more rapidly, on average, than that of the asexual species. But what happens when you factor in the larger growth-rate variability? Which species is more likely to become extinct sooner, and after how long, on average? If you’d like to be more realistic, you can add a random population growth or decrease of up to 10 percent every generation.

If we ignore the last part, as most readers did, the problem is simple, as described by Sayantan Khan and Mark Pearson— if you understand how standard deviations work in the normal distribution, or bell curve. In our case, the growth rate of the populations has a different value each generation, distributed around the mean, with 99.7 percent of these values within three standard deviations of the mean. So for the sexual species, 99.7 of the time the growth rate will be within 0.65 to 1.55, calculated as the mean of 1.1, plus or minus three times the standard deviation (0.15 × 3 = 0.45). In rare cases, the values can range even further away from the mean, and this is where the fatal flaw comes into play when the standard deviation is high: the value can reach zero, which implies extinction. To determine how likely this is you simply determine how many standard deviations the mean is from zero and obtain the area of the normal distributionfrom a standard table or formula, which will give the probability of a zero growth rate.  In the asexual species, the zero growth value is four or more standard deviations away from the mean, in the downward direction — a circumstance that has a probability of 0.000031671, which will occur once in about 31,574 generations on average. (You can calculate this probability in Excel by using the formula =NORM.DIST(0,1.2,0.3,1), or by entering the values in an online normal distribution calculator.)In the sexual species, in contrast, the zero multiplier value is 7.33 or more standard deviations away from the mean, in the downward direction, giving an extinction probability of 1.12249 × 10^-13or once in 9 trillion generations — or in practice, hardly ever. The low number of generations before the catastrophic collapse of asexual populations is predicted by the Red Queen hypothesis, which proposes that organisms must constantly adapt to survive while pitted against ever-evolving opposing organisms, something that asexual creatures cannot do as easily as sexual creatures can because of their extreme similarity and lack of genetic diversity. As the evolutionary biologist John C. Avise from the University of California, Irvine, comments in a recent review on asexual (clonal) reproduction in vertebrates: “Perhaps the maximum well-documented geological age reported for any extant vertebrate clonal line is about 60,000 generations, but in evolutionary terms, this duration is ‘but an evening gone.’” This last poetic touch came from a 1992 Nature article by the famous British evolutionist John Maynard Smith (1920-2004).

The subject of how two sexes arose and have since been maintained in the biological world is endlessly fascinating. Michael Simmons was disappointed that the puzzle did not discuss the question of three sexes more fully and provided some comments of his own, and I agree with them. Three or more sexes would simply add too many complications in biology, with no compelling advantages over what two sexes already provide. As for Simmons’s comment that “many plants, bacteria, algae and fungi are asexual, and they seem fine for it” — that’s true, but most of these types of organisms have found ways to exchange genetic material without sex, or they reproduce sexually in some generations during times of stress. For that reason, I’ve stressed that this analysis only applies to vertebrate animals that are necessarily sexual or asexual. Another factor to consider is the advantage that asexual plants and animals have in spreading and colonizing new territories, a process that a single individual can initiate. This explains the widespread geographic distribution of asexual plants and some all-female asexual animals, such as the Brahminy blind snake, also known as the flowerpot snake because it has colonized almost all the continents by stowing away in the flowerpots of exotic plants sent to faraway destinations. Of course, these asexual species will have a high risk of collapse in “an evening of evolutionary time.”

Todd K. asks the following interesting question about dinosaurs: “Could it not possibly be the case that they were asexually reproducing creatures, and therefore doomed to eventual extinction as described here?” The consensus of expert opinion is that dinosaurs were sexual creatures, based on possible sex differences seen in their skeletons; the fact that their modern descendants, birds, are sexual; and the fact that no large group of vertebrate animals consists entirely of asexual species. They most likely did not die off because of asexuality.

Thank you to all for your interesting comments. The Quanta T-shirt for this month goes to Sayantan Khan. Congratulations! See you again next time for new insights.

Correction: On July 28, 2017, in the first paragraph after Problem 3, a misstatement about the growth rate of the sexual species and how to calculate it was corrected.