Archive for the ‘Math in Life’ Category.

The Age of Consent

Tim Gowers discussed the age of consent on his blog, which I can no longer find. I will talk about his post here based on my old notes and my memory. The age of consent is a legal term to protect young people from being manipulated into agreeing to sex. Having consensual sex with people under the age of consent may be considered statutory rape or child sexual abuse.

Gowers starts with several assumptions.

  • Non-triviality: There should exist an age at which a person is qualified to consent to sex and, consequently, have it.
  • Simplicity: Whether or not two people are allowed to have sex with each other should depend only on their age in years.
  • Monotonicity: If two people are allowed to have sex with each other today, they should be allowed to have sex with each other at all times in the future.

From these assumptions, the following theorem can be deduced.

Theorem. The only possible rule satisfying these assumptions would allow any two people to have sex with each other as long as they both reached some fixed age k.

There is a problem with this type of rule. Suppose k is 18. If two people who are slightly younger than 18 have consensual sex, they can’t both be predators. These are two children with raging hormones. There is no reason to punish anyone. Now imagine that one of the partners turns 18. Society would still consider this a Romeo-and-Juliet case and would tend not to punish such a partner. Now imagine a child younger than 18 having sex with a partner over 40. The older partner has no raging hormones, knows what they are doing, and probably knows how to manipulate little children into having sex. So, it might be desirable to have a rule that differentiates between these two cases. The rule would take into account the difference in ages while forgiving younger offenders and still punishing predators.

Consider the most common type of law to resolve this issue: Anyone older than 18 can have sex, and, in addition, a person who is not older than 20 can have sex with someone between the ages of 16 and 18. This law doesn’t satisfy monotonicity. It could be that one day the older partner is not yet 20, and the next day, oops, they have a birthday. So, as a birthday gift, they are not allowed to have sex with each other anymore.

Here is a simple idea to resolve the issue by having the law focus on the age gap instead of the age of the older partner. We can have an adjusted law: Anyone older than 18 can have sex, and, in addition, a person can have sex with someone between the ages of 16 and 18 as long as the age gap is not more than four years. This rule doesn’t satisfy the simplicity assumption above, but it is simple enough. It is close in spirit to the previous rule and satisfies monotonicity. The problem with this rule is continuity.

  • Continuity: If the age gap between couple A is only slightly larger than the age gap between couple B, then couple A should not have to wait significantly longer to be allowed to have sex.

According to the adjusted rule, the couple with the age gap of four years and one day might have to wait two years longer to have sex than the couple with the age gap of four years. This seems unfair.

Tim Gowers suggests dropping the simplicity rule. We can use days rather than years. For example, the rule might be that if one person in a couple is under 18, but at least 16, and has age x, then the other partner has to be not more than age y, where for example, yx = 4 + (x − 16)/2. So when one partner turns 16, their partner has to be not older than 20. When one partner is 16 and two months, the other cannot be older than 20 and three months. With the younger partner getting older, the allowable age gap is increasing slowly. By the time the younger partner is a day from turning 18, their partner can be almost five years older.

It might be complicated for two people to calculate if they are allowed to have sex according to this formula. But Gowers’ big idea was that apps and websites could do this easily: two people plug in their birthdays and know whether they are allowed to have sex.

Share:Facebooktwitterredditpinterestlinkedinmail

The Problem with Two Girls

Puzzle. Two girls were born to the same mother, at the same time, on the same day, in the same month, in the same year, and yet somehow they’re not twins. Why not?

I won’t tell you the expected answer, but my students are inventive. They suggested all sorts of scenarios.

Scenario 1. There are two different fathers. I had to google this and discovered that, indeed, it is possible. This phenomenon is called heteropaternal superfecundation. It happens when two of a woman’s eggs are fertilized by sperm from two different men. Unfortunately for my students, such babies would still be called twins.

Scenario 2. The girls are born on the same date, but not on the same day. This could happen when transitioning from the Julian to Gregorian calendar. The difference in birth times could be up to two weeks. I had to google this and discovered that twins can be born months apart. The record holders have a condition called uterus didelphys, which means that the mother has two wombs. Unfortunately for my students, such babies would still be called twins.

Scenario 3. The second girl is a clone. This scenario can potentially happen in the future. Fortunately for that student, I suspect that such babies would be called clones, not twins.

I decided to invent my own scenario outside of the actual answer, and I did.

Scenario 4. Two girls are from the same surrogate mother, but they are not twins. I had to google this and discovered that this actually happened: Surrogate mother of ‘twins’ finds one is hers.

Sometimes life is more interesting than math puzzles.

Share:Facebooktwitterredditpinterestlinkedinmail

Ready for My Knot Theory Class

Ready for Knot Theory Class

I used to hate crocheting. Now it’s been growing on me.


Share:Facebooktwitterredditpinterestlinkedinmail

Trying to Crochet the Impossible

Hyperbolic Surface trying to fit

I’ve been crocheting hyperbolic surfaces of constant curvature. The process is time-consuming, so while I am crocheting, I wonder about the mathematics of crocheting.

Hilbert’s theorem says that I can’t embed a hyperbolic plane in 3-dimensional space. The proof is rather involved. But here, I have an explanation from the point of view of a crochet hook. My hook starts with a tiny cycle of four stitches. Then for every x stitches the hook makes y stitches in the next row, where y is greater than x. The extra stitches should be evenly distributed to guarantee that locally every small area is approximately isomorphic to other areas, meaning that the surface has a constant curvature.

The ratio of stitches in the next row to the current row is r = y/x. Thus, the number of stitches in each row increases exponentially. But each row is a fixed height h. That means after k rows, my thingy has to fit inside a ball of radius kh. But the length of the last row is 4rk-1. It becomes huge very fast. As the last row is a physical curve made out of stitches, there is a limit of how much of it I can fit into a given volume, creating a contradiction.

That means, if I start crocheting, something should happen that won’t allow me to continue. I decided to experiment and see what actually would happen. Being lazy, I preferred the disaster to happen sooner rather than later. So I chose the ratio of three: for each stitch on my perimeter, I added three new stitches. Shortly after I started to work, the process became more and more difficult. The ball was too tight. It was challenging to hold that thing in the place where I needed to insert the hook. And the loops were getting tighter, making it more exhausting to insert the hook into the proper hole. So each new stitch was taking more and more time to complete.

To my disappointment, the thing didn’t explode, as I was secretly hoping: I just couldn’t work on it anymore.

Share:Facebooktwitterredditpinterestlinkedinmail

The Stable Marriage Problem and Sudoku

As you may know, I run PRIMES STEP, a local program where we do mathematical research with students in grades 6-9. Last academic year, we looked at the stable marriage problem and discovered its connection to Sudoku. Our paper The Stable Matching Problem and Sudoku (written jointly with Matvey Borodin, Eric Chen, Aidan Duncan, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, Michael Voigt) is now available at the arxiv.

Consider 3 men and 3 women who want to be married to each other in heterosexual couples. They rank each other without ties. The resulting 6 permutations of numbers 1, 2, and 3 that describe the six rankings are called the preference profile of this group of people. A matching is unstable if two people would be happier to run away together than to marry into the assigned couples. The two potential runaways are called a rogue couple. A matching is called stable if no rogue couple exists. The Gale-Shapley algorithm, invented by Gale and Shapley, finds a stable matching for any preference profile, implying that stable matching is always possible.

We discovered that preference profiles form a natural bijection with ways to place one digit into a Sudoku grid. So we wrote a paper describing the stable marriage, rogue couples, the Gale-Shapley algorithm, soulmates, and such in terms of Sudoku.

Oops, I forgot to explain who the soulmates are. We invented this term to describe two people who rank each other first. Though it is possible to have several stable matchings for the same preference profile if the soulmates exist, they must always be matched together.

We also invented a new Sudoku type, which I will explain next time.

Share:Facebooktwitterredditpinterestlinkedinmail

Coronavirus and Gender

You probably heard in the news that more men are dying from coronavirus than women. But not in Massachusetts. Here the proportion of women is about 52 percent. Why is this the case? Being a woman, should I be worried that I live in Massachusetts?

We know that coronavirus strikes older people harder than younger ones. Thus, we should take age into account. In the US more boys are born than girls. By the age of 40 the ratio evens out. Starting from 40 there are more women than men. With each next age group, the disparity increases. According to a recent US population report and for ages 85 and over there are about 4.22 million women versus 2.33 men: the proportion is almost 2 to 1.

As the coronavirus targets older people, were it gender-neutral, we would have had way more female deaths than male. This is not the case. So it hits males harder than females. But why are the ratios of female to male deaths different for different countries and states?

One simple explanation is that this is related to life expectancy and the age of the population. The older the population, the bigger the percentage of females. Which in turn increases the proportion of female deaths.

It could also be that Massachusetts has good health care making the average age of dying patients older than the average age for the country. This in turn will increase the proportion of females dying from coronavirus. No, I am not worried about living in Massachusetts.

Share:Facebooktwitterredditpinterestlinkedinmail

Coronavirus in NYC

It was reported last week that that 37 NYPD members died of covid-19. I assume that they were way below 65. It is known that the coronovirus death rate for people below 65 is a quarter of the total death rate. That means, 37 people in NYPD correspond to at least 150 people in general. Assuming that the mortality rate of coronavirus is 1 percent, the number of infected NYPD members a month ago was 15000.

By now, it could be that more than half of NYPD was infected.

NYPD members have to communicate with people a lot due to the nature of their work. That means they are more prone to being infected. At the same time, they transmit more than people in many other professions.

I can conclude, that about half of the people that are high transmitters in NY have antibodies by now. Assuming they are immune, the covid transmission rate in NY has to be down.

Assuming the immunity stays with people for a while, the second wave in NY can’t be as bad as the first one.

Share:Facebooktwitterredditpinterestlinkedinmail

US Coronavirus Numbers

Every day I check coronavirus numbers in the US. Right now the number of deaths is 288 and the number of recovered is 171. More people died than recovered. If you are scared about the mortality rate, I can calm you and myself down: our government is incompetent—the testing wasn’t happening—that means the numbers do not show people who had mild symptoms and recovered. The real number of recovered people should be much higher.

Scientists estimated the mortality rate of coronavirus as being between 1 and 3.5 percent. Also, they say that it usually takes three weeks to die. That means three weeks ago the number of infected people in the US was between 8,000 and 29,000. The official number of cases three weeks ago was 68. I am panicking again—our government is incompetent—three weeks ago they detected between 0.25 and 1 percent of coronavirus cases. If this trend continues, then the official 19,383 infected people as of today means, in reality, somewhere between 2 million and 8 million infected people.

I can calm you and myself down: the testing picked up pace. This means, the ratio of detected cases should be more than 1 percent today. Probably the number of infected people today in the US is much less than 8 million. I am not calm.

Share:Facebooktwitterredditpinterestlinkedinmail

Less Annoying Hyperbolic Surfaces

Less Annoying Hyperbolic Surfaces

I already wrote about my first experience crocheting hyperbolic surfaces. In my first surface I added two more stitches per current stitch. It took me hours to crochet the last row: the same hours it took me to crochet the rest.

For my next project, I reduced the ratio. The light blue thingy has ratio 3/2. I continued making my life simpler. The next project, the purple surface on the left, has ratio 4/3. The last project on the right has a ratio of 5/4 and is my favorite. Mostly because I am lazy and it was the fastest to make.


Share:Facebooktwitterredditpinterestlinkedinmail

Happy 2019!

Happy 2019, the first 4 digit number to appear 6 times in the decimal expansion of Pi.

By the way:

2019 = 14 + 24 + 34 + 54 + 64.

Also, 2019 is the product of two primes 3 and 673. The sum of these two prime factors is a square.

This is not all that is interesting about factors of 2019. Every concatenation of these two prime factors is prime. Even more unusual, 2019 is the largest known composite number such that every concatenation of its prime factors is prime. [Oops, the last statement is wrong, Jan 3,2019]

Happy Happy-go-Lucky year, as 2019 is a Happy-go-Lucky number: the number that is both Happy and Lucky.

In case you are wondering, here is the definition of Happy numbers: One can take the sum of the squares of the digits of a number. Those numbers are Happy for which iterating this operation eventually leads to 1.

In case you are wondering, to build the Lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, …. The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, …. The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, …. The next number remaining is 9, so delete every ninth number, etc.

Share:Facebooktwitterredditpinterestlinkedinmail