The Magical Math of Magicicadas

The Magical Math of Magicicadas

This post was originally published in 2014 and has been update to celebrate the emergence of Brood X in Maryland, USA in the spring of 2021.

Autumn leaves drift down, silently ushering the chilly advent of fall. Gone are the noisy days of summer, synonymous with the incessant, and insistent, chirp of the cicadas made by rapid vibration of abdominal tymbals and orchestrated by a frenzied mass of mating males. Did you know that the chirp of a cicada clocks in at 120 decibels, enough to cause permanent hearing loss in humans?!

A Plague of Primes: Periodical cicadas, of the North American genus Magicicada, have a bizarre life cycle, spending 13 or 17 years underground as immature nymphs, emerging briefly to live, love and die as adults.

Seventeen years of peaceful dreaming,

Followed by a week of screaming*.

Their coordinated emergence, triggered when the soil warms to precisely 64 degrees F, guarantees a plague of biblical proportions: the densest broods can number 1,000 cicadas per square meter! Is there a mathematical basis for 13 or 17 year life cycles? You may have noticed that both are prime numbers: divisible only by themselves and the number 1.

Mathemagics: In the computer simulation** graphed in the image above, notice that 13 and 17 year periods produce the most survivors. The cicadas only defense against predators is their sheer number, and their survival strategy is simple- predator satiation. The prime numbers work better because they decrease the chance that the life cycle of the cicada matches that of its predators. A 12 year life cycle, in contrast, is a particularly bad choice: predators that reproduce every 2, 4, or 6 years (all divisors of 12) would feast on the hapless cicadas. Hiding underground for long periods helps survival, but reproduction cycles that are too long may result in being out competed by other species. Shorter prime numbered cycles may be weeded out if co-emergence of different broods results in hybridization and altered life cycles in the offspring.

Allee Effect: Biologists refer to the penalty of small population size on individual fitness as Allee Effect, named after W.C. Allee who showed, in 1932, that goldfish survived better in larger populations. The Allee effect means that there is a critical population size, below which the population becomes extinct. If the Allee effect is applied to simulations of cicada populations, successful cycles are in the order 17> 13 >> 19 year cycles, all others become extinct. Without the Allee effect, all brood cycles survive (see Fig. 1 of Tanaka et al., cited below). 

Cicadian Clock: Underground, periodical cicadas undergo numerous molting stages known as instars, emerging at their 5th instar before molting for the last time into the adult form. We know that they time their emergence when soil temperature reaches 64 degrees F, but how do they track the years? It’s believed they have an internal “molecular clock” which keeps time based on changes in the tree sap they consume. Here’s a clue: in 2007 a warm January in Cincinnati was followed by a hard freeze in February, and then a normal spring. As a result, trees produced two sets of leaves that year adding up to 17 leaf sets in 16 years. Cicadas feeding underground on those trees emerged a year early!

So the next time you hear the chirp of the cicada, take a moment to appreciate the simple maths hidden within their lives! 

*“A Cicada’s Life” by Alan Rubin 

**Graph and blog:

#OpenAccess Ref: Allee effect in the selection for prime-numbered cycles in periodical cicadas. Tanaka et al., 2009 PNAS.


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79 Responses to The Magical Math of Magicicadas

  1. Rajini Rao says:

    Thex Dar I can’t imagine summer without them! Turns out there are annual cicadas, that are noisy and plentiful enough, and then there are the periodical cicadas that are something else altogether! 

  2. Rajini Rao says:

    Okay, here’s what I dug up on the name “jar fly”: One is that when you catch one and hold it in your hand it “jars” or vibrates. The other thought is that the nickname came from the constant singing that might “jar” or unsettle some people’s nerves who are not accustomed to hearing it for hours on end.

  3. Marta Rauch says:

    Rajini Rao interesting! Thanks for sharing.

  4. Rajini Rao says:

    Owen McNamara grasshoppers? locusts? 

  5. I wish I could generate atleast 20 decibels with my abdomen. I didn’t have to live with this 😦

  6. Rajini Rao says:

    Haha, there may be better ways to woo the female of our species R Prakash Prakash 🙂

  7. Rajini Rao says:

    Owen McNamara locusts indeed are the stuff of biblical plagues and most people in N. America don’t refer to cicadas as locusts. 

  8. Jesse H says:

    The ones around here come out every 10 or more years. I was lucky enough to be camped out 2 days before they emerged. What a great experience.

  9. Rajini Rao says:

    Owen McNamara the post is more than the drawing, but thanks for pointing that out. Do you approve of this drawing? 

    How about the science behind the Allee effect? Or Predator Satiation? Let’s talk about that. 

  10. Rajini Rao says:

    Jesse H the one outbreak I still recall, people were frying and eating them. Also, selling chocolate covered critters :O

  11. This is a beautiful post! Thanks for sharing! Love the co relation between primes and survival

  12. Kawthar A says:

    Great read as always, Rajini! Hello btw 🙂

  13. Rajini Rao says:

    Roerich Warton thanks- this post is in response to a comment I came across in one of the science communities, where someone said “maths is not science”. Purely theoretical maths is not science, I understand, but mathematics underlies so much of our natural world. 

  14. Chad Haney says:

    Here’s an interesting tidbit, cicadas use their singing muscles in their abdomen to generate heat.

  15. Rajini Rao says:

    Oh hello, Kawthar A ! I wouldn’t imagine you would like some cicadas, even if Nutella or chocolate covered? 🙂

  16. Kawthar A says:

    Heheheh…I’m full of surprises :p Rajini Rao

  17. Rajini Rao says:

    Chad Haney I had not heard of that tidbit! Apparently, body temps can rise up to 17 degrees C over ambient:

  18. Rajini Rao that’s one of the reasons I stay away from most of the scientific communities, the focus is so much on the details they forget to appreciate the wonder that exists in nature.

  19. Chocolate covered cicadas? 😮

  20. Pam Adger says:

    Rajini Rao yes they have cookbooks. Eating and experiencing them is on my bucket list.

  21. Rajini Rao says:

    Pam Adger do report back if you try them. #tasteslikechicken  😀

  22. Chad Haney says:

    For a little Sunday comic fun, Matthew Inman shows how honey bees use heat to defeat wasps.

  23. Rajini Rao says:

    Chad Haney he’s cartooned it, great! (The video was creepy, Oatmeal’s version is less intimidating.) 

  24. Rajini Rao says:

    Rashid Moore apparently there is now a Doritos flavored Mountain Dew. So, cicada drinks up next? 

  25. Rajini Rao says:

    Haha, thank you kindly Rashid Moore ! I’m game for trying the vegan version. Chickpea-kayda sounds like pakodas 🙂

  26. Rajini Rao says:

    Sounds just right 😛

  27. Jim Carver says:

    In Texas most people call them locusts or katy-dids. There are the annual types, but every 13 or 17 years (not sure) we get the LOUD ones. It’s surprising how something that small can make so much noise.

    Btw, 13 is also a Fibonacci number: 1,1,2,3,5,8,13…

    And yes, the graphic is wrong, those are certainly grasshoppers. Why is it important? Because some kid will see that and get the wrong idea.

  28. Real interesting … TIL 🙂

  29. I have never heard or eaten a cicada in my life, maybe some day I will do both.

    But it reminds me that I heard many stories about them, some of the best, of course here:

    Caustic Soda, in case you do not know them, you can thank me later.

  30. Jim Carver says:

    abhishek chamaria They sound like if you shrunk a chain saw down to 100th of its original size and kept it just as loud.

    I don’t know what they taste like either. Probably similar to crickets. I know there’s nothing left of them after they die, merely a shell that you can crush into dust.

  31. Deen Abiola says:

    There’s a piece that’s missing when I reason backwards. Naively, you’d expect population to remain close to a steady state but what you’re saying is that population actually oscillates meaningfully with a period close to reproductive cycle lengths. That a significant number all mate close to the same time?

    Another observation: why don’t the predators also shift to mating at the same cycle? In a single predator prey relationship this leads also to a die off so a strategy that is off prime allows replenishment of prey. This means the cicadas remain so because the predators eat other things otherwise you’d expect, after prey extinction, that eventually, they’d spread out from this rather strict strategy. 

    All in all I’m not sure that the case for an exploitation of primes is sufficient as an explanation. 

  32. Jim Carver says:

    Deen Abiola Not sure I understand all that jargon and theory, but what they do is breed infrequently in great numbers so that predators do not have a large population that is accustomed to eating them. So when they are in their ‘on’ year, they basically go unchecked. Since they go to sleep after that, no predator can make a living on them.

    I don’t know the biology terms to say that scientifically.

  33. Rajini Rao says:

    Deen Abiola the entire population mates within a week or two, after which all the adults die en masse. Cicadas mate only upon molting into adults, after they emerge from the ground. Their emergence is highly coordinated and their sheer number overwhelms their predators.

    There are no predators who have evolved to eat only (or predominantly) cicadas. So they go about their business reproducing in response to their own specific set of biological requirements and selection pressures. Instead, the predators are opportunistic, eating the cicadas when they show up. The cicadas outcompete them only by their numbers. 

    You may find this figure interesting:

  34. Rajini Rao says:

    Jim Carver thanks for that explanation. Only minor point is that the immature nymphs are actually growing and developing underground, feeding on saps of roots. 

  35. Jim Carver says:

    Rajini Rao That’s what it looks like when a biologist says it. 🙂 I could have worked on mine more and made it more biology-y.

    It’s a pretty amazing strategy and you have to think how long it took to come up with this. All the little details have to be just right. Amazing. Of course the really amazing part is time. It took a lot of time to come up with it, but there was a lot of time to give.

  36. Rajini Rao says:

    Jim Carver it is amazing. I’m glad I stumbled on to the topic in response to someone in the science community stating that “maths isn’t science” 🙂

  37. Jim Carver says:

    Rajini Rao Well, hmm, I guess it could be true semantically, but isn’t that like saying “speech isn’t language”?

  38. Rajini Rao says:

    Right, there are some aspects of theoretical math that are purely hypothetical, not limited by reality, and other branches that are essential to explain the natural world around us. 

  39. Jim Carver says:

    Rajini Rao I was always confounded by things in mathematics that are natural, such as Fibonacci numbers and ratios, natural logs and the imaginary set i . I think those things are going to come out no matter what system you use.

    We seem to go through cycles alternating between highly complex and quite simple. We think things are highly complicated and find a unifying principle, then things get more complex and we find there’s another simpler way…

    Did you know there’s an “organic’ theory in physics that has some support?

     They postulate that if matter and energy were made up of string particles, they could form sequences just like what we see in DNA. Perhaps only four types of strings, depending on how they were arranged, could account for all types of particles/waves.

    They say the math is behind them. I think I’ll have to take their word on that though. 🙂

  40. John Kampsen says:

    Rajini Rao Fine post, my friend. Permanent hearing loss ?

    That’s some loud insects !

  41. Rajini Rao says:

    Thanks, John Kampsen . Hearing loss occurs if the chirp is right next to the ear, apparently. At a distance, it’s just a cicadian cacophony 🙂

  42. Good post Rajini Rao; as Owen McNamara mentioned, I also thought they looked like grasshopers, which is what I remember from my childhood.  Here’s a link with a photo of cicadas –

  43. Jim Carver says:

    John Kampsen 

    Yeah, it gets pretty bad, one summer when I was a teenager we had them. It was constant even with the windows closed. That summer was stressful and we were really glad when winter came. They let up a little overnight, but as soon as the sun comes up, they’re at it again. You can spray water but it only lasts about 30 seconds. Horticultural oil would kill the ones in your area, but it wouldn’t do anything because the ones not sprayed from adjacent areas would re-invade quickly because there are so many. It’s kind of easy to see why some people call them ‘locusts’. But I’ve never seen them to cause any appreciable damage, so the name is not a very good one.

  44. Deen Abiola says:

    Rajini Rao No, I don’t mean the cicadas — I mean the predators. In order for the predator reproductive cycle to have any kind of significant effect wrt cicada emergence then most of the predators will have to be mating around the same time.

    You know, on the timeline birth and death event’s would have to be clustered instead of uniformly distributed. And a tight lifespan distribution too.

  45. Jim Carver says:

    Deen Abiola I don’t know how well you are following, but consider that the predators would also have to have the same life cycle, and that is extraordinarily unlikely, the chances of me spontaneously combusting on the spot are greater. 

    Don’t you get it? They reproduce every 13 or 17 years, not every year, that number of years! Now try and understand predator/prey relationships in your mind and stop thinking in terms of silly formulas. This is not the time for formulas.

  46. Deen Abiola says:

    I follow perfectly well. What I’m saying is that Gould’s guess is backed mainly by simplistic Lotka-Volterra style computational models. The assumptions involved are strong; they typically only involve paired population predator prey dynamics and smooth population life cycles. Things aren’t so neat in reality. The models are great for getting broad stroke ideas of what’s going on in various competitive scenarios but aren’t strong enough to justify conclusions of this strength.

    I’m told, Cicadas are born every 17 years to avoid being feasted on when they’re out in force. Ok. Why exactly is that better than every 12 years? Because animals with 1,2,3,4,6,12 year cycles would feast on them otherwise. So, this is telling me that animal populations follow a cycle strong enough to exert sufficient selective pressure. Cicadas are born when other species populations are less, implying populations spike. So spikes exist and are correlated with reproductive cycles. Working back from that you get to my questions and disagreements above.

    And what about with scenarios more complex than a simple predator prey relationship; even assuming spikes, what if there were enough species such that one was always occurring every year? Or consider this simpler example: Cicada has 13 year cycle but the predator has an 8 year cycle that started 5 years after the cicada. The point here is to emphasize the assumptions: spiking cycles, pair dynamics, alignment wrt introduction, tight lifespan, clustered mating and the assumption that years instead of months or weeks are the most meaningful time unit when it comes to population dynamics (are they?).

    I’m not disputing the notion that there might be something significant about having settled on primes; instead I’m questioning the idea that avoiding being feasted on has sufficient explanatory power.

  47. Jim Carver one of the things I like the most about being in Texas is the locusts. I love to hear the buzzing. One of the things I do not like are the locust killer wasps we were dive bombed by them, couldn’t sit on the deck or walk to the pool without harassment. They were frightening before we figured out what they were. It’s been a couple of summers and I plan on laying new landscape cloth down to trap the emerging wasps when they come our to feast on 2 or 4 year locusts next summer. 

  48. Rajini Rao says:

    Deen Abiola in your example of a predator whose 8 yr life cycle started 5 yrs after the 13 yr periodical cicada, the predator and prey coincide and then diverge in subsequent cycles. This means that the cicada has the chance to re-establish population after taking a hit. The simulations are done over thousands of years, as you can see in this figure:

    The other hypothesis that explains the origin of long periodical cycles has to do with the cold summers of the Pleistocene era, when the cicadas arose. Because this genus needs a minimum temperature to reproduce, longer life cycles protected them from the vagaries of the summers. See the nice explanation in this news story:

  49. Deen Abiola says:

    Thanks for the link to an alternative, Rajini Rao but the key point of all my examples was too show that the choice of parameters strongly controls whether it’s a coincidence or compelling; with regards to the simulation starts, the underlying dynamics for divergence has more to do with feedback between population sizes than it does something fundamental about where in the start the cycle was.

    Looking on Google Scholar it seems I’m not the first by far to hold these misgivings. Unfortunately there doesn’t seem to be any compelling nor wildly accepted explanation. The pleistocene era one seems to be undermined by genetic evidence suggesting an earlier onset of this cycle.

  50. Rajini Rao says:

    Deen Abiola “the choice of parameters strongly controls whether it’s a coincidence or compelling”. Actually, no. As the outcomes in the PNAS paper demonstrate: “The various parameters in our model seem to have little effect on the eventual outcome; under a wide range of extinction thresholds, non-prime life cycles are unlikely to persist (Figs. 1​1​​–5). In addition, the numerical advantages of prime-numbered cycles appear only when an Allee effect is at work (Fig. 3). These results suggest that the mechanism of prime-number selection is extremely stable under various environmental settings.” Your wording is a bit vague, so it would be helpful to me if you were more specific. For example when you brought up the 8 yr predator vs. 13 yr cicada cycle, that was a straightforward issue to address. 

    The life cycle divergence in Magicicada (13 vs. 17 yrs) is closely associated with global climatic fluctuations and shorter growing seasons in the north (17 yr cycles) versus the south (13 yr cycles). Based on phylogenetic analysis, it appears there is a common genetic basis for the two life cycles (dating to 3.9 Mya), which are shared by 3 groups of Magicicada species. The two prime numbered cycles can shift within groups by invading cicada species, followed by synchronization with the resident populations due to natural selection. There is a discussion of this here:

  51. Deen Abiola says:

    By parameters I mean not just the parameters of the model but also the choice of what to model in the first place! For example models with two species will seem to have stable parameters but you still have the notion of period length embedded in the model and according to above, there are advantages only when factors are placed in to model the Allee effect. That the model works for the studied scenario only strengthens the probability of the model not the converse. But what about a more complex scenario with population numbers that do not change much, or if not: multiple predator species undergoing cycles as to undermine a cyclic advantage to predator avoidance? Or periods in predator number that are not fix but fluctuating so hard to predict (by evolution)?

    Models that are dynamical will not be dominated by when “starts” occur in the timeline (indeed such a thing would make no sense, my example was not dynamical but more to emphasize that what you choose to look at is important; in this case, initial start vs coincident start in a simple linear model makes a difference. Another: months vs years)

  52. Deen Abiola says:

    In The priming of periodical cicada life cycles they argue :>Periodic cycles evolved much earlier, and were independent of glacial effects if the rate of mtDNA divergence has been correctly estimated.  Indeed, they can be accounted for solely in terms of biological interactions, including interspecific competition, without recourse to explanations involving temperature change, even though variable weather could have had modifying effects on life-cycle evolution. 

  53. Deen Abiola says:

    The view point I most favor is something like the niche specialization as found in Galapagos Finches. Competition for resources by various cicada species (not all of them are periodic) being the stronger driver. 

  54. Deen Abiola says:

    One more thing — and sorry to go on so. When I saw the explanation of predation to be the driver, it seemed too neat, tasting too much like a just so hypothesis. My issue was based mainly with unacknowledged assumptions (parameters) baked into the models. In particular, one of not just a single other predator but one with any kind of easily discernable and fixed population cycling. 

    A quick survey of articles confirmed my suspicion in that there is no clearly or widely acceptable explanation. In this article postulating on how the counting might be done, the issues I noted are raised near the end:

    > Whether the cyclic predator species exist remains an open question. And some quite different explanations have also been put forward. For example, Randel Tom Cox of Arkansas State University and C. E. Carlton of Louisiana State University argue that the heart of the matter is not predation but hybridization. Interbreeding between broods that differ in period could disrupt synchronization for both groups. Thus 13–year and 17–year broods are favored because they emerge together only once every 221 years.

    And in the too neat observation I am decades late:

    > In the 1960s Monte Lloyd of the University of Chicago and Henry S. Dybas of the Field Museum of Natural History in Chicago offered a curious meta–theory of cicada evolution. Any theory that seems too plausible, they argued, is automatically suspect. “If there were a broad, easy evolutionary highway towards periodicity, then why would not more species have taken it?”

    I favor explanations based on resources. They’re simpler and neater from requires fewer moving pieces. The only prerequisite being the strategy of oversatiation evolving first. Then everything resource related follows: avoidance of hybridization to maintain synchronicity. And against other aperiodic cicada species competing for the same resources (where cycling population sizes is less tenuous nor does it need any kind of explicit pair dynamic) would also bolster selection for a prime numbered range:

    >To this scheme for the initial evolution of periodicity, I suggest adding an overlooked factor, that of competition with the nymphs of other species of non-periodical cicadas. Interspecific competition would augment the effects of the between-brood component of intraspecific competition in Bulmer’s model [1], thereby making the evolution of periodicity more probable. 

    Interspecific competition would have been a significant factor if the ancestral species of Magicicada had invaded forested habitat, already occupied at high densities by one or more other species (e.g. Tibicen spp. ), from another habitat such as adjacent prairie grassland


    >cicadas of all other species (perhaps 3000 worldwide) are not synchronized, so some adults mature each summer and emerge while the rest of the population continues to develop underground. Many people refer to these non-periodical species as annual cicadas since some are seen every summer. The life cycles of most annual species range from two to ten years, although some could be longer.

    These hypotheses better avoid the issue of why not others because the strategy of periodic emergence with over-satiation is rare and so strategies around synchronization, and avoiding hybridization — that also have the beneficial corollary of (mildly) ameliorating resource competition — rarely have all the correct components to be beneficial. 

  55. Rajini Rao says:

    OMG, Deen Abiola that’s quite a wall of text. I’m just done with work, but I will cite a couple of problems right off with your shortest comment on resource competition as with finches (I’ll read the other comments  in the morning!). First, resource competition is not a problem with adult cicadas. Unlike locusts, cicadas do not consume much natural resources ..they mate, lay eggs and die. The nymphs burrow into the soil where they are xylem feeders, consuming mostly water from roots (xylem is 99.9% water; this energy poor diet contributes to their very long time to maturation). Second, resource competition as in finches does not explain prime number and long life cycles. 

  56. Deen Abiola says:

    Heh, I didn’t write about 60% of it =) Most of it is quotations from a paper and article, both of which contradict your above post though. Prime number aspect is also addressed.

  57. I just returned from a visit back east (where I lived for decades)…. and realized then that I do indeed miss the sounds of the night. Especially in the forests….thanks for the reminder. Rajini Rao 🙂

  58. Jim Carver says:

    “Droner…you know you are a droner…when you put your hands in your head, oh no!. I said, ” Far Out! ” , a day, a life to live. You know you had it comin’ to you. Now, there’s not, at lot I can do.”

  59. Mary T says:

    Lovely and fascinating post Rajini Rao ~ I don’t think I have ever heard a cicada.  The mathematical aspect is quite something!

  60. Chad Haney says:

    Mara Rose​ you might want to hear one cicada but not a few thousand.

  61. Mary T says:

    Good point, Chad Haney, lol :-).

  62. Coming very late to the discussions.

    First, Rajini Rao, great post and thoughtful.

    If the standard deviations in model created over the long cycles match the population standard deviations in real life then mostly the model would hold good for larger thesis.

    The prime numbers being what they are would have very direct correlations to any predator life cycle overlaps.

    There is no particular need to suspect it.

    I will only reduce the importance of life cycles synchronicity IFF (if and only if) there exist a large number of predators with yearly cycles or very very large number of predators with base 2, 3 cycles.

    The reason is, those above predators have significant probability to coincide with 13 years cycles by life cycle standard deviations of small cycle predators if the models are run for very large time set.

    ON another topic-Rajini Rao

    At times, it looks to me that natural selection and evolutions are very very inefficient. There has to be so many variations continuously happening all the times.

    What always bothered me, is a variation just found unsuitable can not be discarded totally. The same variation may be very suitable say 20000 years from now.

    So, the life cycle being 13 years may be great so far but may be that may prove to be deadly once.

    The point is, evolutionary forces are game of win at a point in time. It does not say much about tomorrow.

  63. Rajini Rao says:

    Very interesting analysis, mandar khadilkar , thanks! In the case of a large number of predators with yearly cycles or those with base 2,3 cycles, I’m guessing that these surely exist so that the cicada population has already adapted to saturate their appetite.

    For fun, you can press start and run the simulation on this site:

    Your point about evolution is spot on! As consolation, I would point out that mutations arise by chance and the same (or similar) mutation may well appear again in the future for another go at sending us down a different path. We do see the same adaptation appearing independently in different species at different times,. A grand game of chance 🙂  

  64. Dorothy Pugh says:

    Very interesting and well-supported argument for avoidance of predators! What’s also interesting is that our 13-year cicadas (and the 17-year cicadas a little north of us) came for a month in late spring, entirely missing the hot-weather dog-day cicadas, which start emerging on the first day it gets up to about 95F that year.  That way they avoid the cicada killer wasps and other cicada predators that come out every summer.

  65. Rajini Rao says:

    Dorothy Pugh it’s interesting that the periodic cicadas do not coincide with the annual cicadas- I had not noticed that! 

  66. Mumtaz hussain $ paris $

  67. Ted Pavlic says:

    Nice write-up. But why are orthopterans depicted around the graph? Cicadas are not true locusts.

  68. Rajini Rao says:

    Ted Pavlic yes, somewhere in this long comment thread we did go over this 🙂

    I took the image from the math blog that I referenced; should’ve known that biology would not have been their strong point. 

  69. Ted Pavlic says:

    Rajini Rao – Gotcha. 🙂 That being said, good mathematical biologists understand their model system and related phylogeny and natural history. Unfortunately, it’s very easy to get by without taking the time to learn about what you’re studying. Hopefully in the future with the emphasis on interdisciplinary research, there will be less of this sort of silliness. 🙂 

  70. lx74 shri says:

    Had how ? How is this

  71. Gary Klafta says:


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