Celebrity Marriage Math

In this blog we have talked about the surprising math of cooking, relieving math of parenting, curious math of travel and helpful math of dating. Yes, this is correct - dating. What to do to increase the chances of accidentally bumping into the subject of your affection, how to count his exes and when to settle down confident in your choice. Now - when we talk dating and marriage there is nothing more magnetizing than celebrity coupling. Who went to Oscars with whom, what were they wearing, how long they are dating, their age differences and whether they will be together the next year?

A new formula published this week in the NYTimes claims to predict the stability of celebrity marriages. This formula has been created by Garth Sundem and John Tierney six years ago and recently refined by Sundem based on the plenitude of the new scandalous data.

Before revealing the actual formula, let's think what public variables one could use to come up with a good couple compatibility predictor. Probably ages and age differences, number of past marriages and their length, number of years couple dated before marriage. There should a time factor - as probability slowly descends with years, although not necessarily linearly.  For celebrity marriages specifically there may be relative fame: when each person in a married celebrity couple spends most of the time away alone shooting a movie or performing they are more likely to start fancying someone else. There is also a type of fame: do you read about them in the NYTimes or mostly in National Enquirer.

Well, we were quite close with most of the variables, but the actual formula is far from simple:

Red variables describe the wife, blue - the husband and gray Md and orange T describe their relationships. Couples combined age (Ah + Aw) is used as age makes us wiser and data shows that younger couples divorce sooner.

Garth discovered that "tabloid fame dooms" celebrity marriages (or, to my opinion, perhaps just reflects their natural disintegration) and it is wife's fame specifically that matters most: ratio of the times she was mentioned in the NYT to ENQ.

What about this strange Sc (aka Sex Symbol) variable? Authors are citating  psychological research  that shows that women who wear sexually provocative clothing tend to be more narcissistic, and narcissism is a good predictor of sexual infidelity.

When you plug into this formula Kate Middleton and Prince William their chances of celebrating 15th marriage anniversary are pretty high - 71%, but for Tom Cruise and Katie Holmes is it only 10%.  Enjoy the full NYTimes article and more couple comparisons here.

February the 29th

February is usually the cold-dark-and-sneeze month that we all wish will end soon, but this year it got extended by one extra day. Leap year. 2012 is divisible by 4 (I checked: 2012 / 2 = 1006, 1006/2 = 503), so here we have February the 29th.  I vaguely remembered that this extra day comes because earth rotation around the sun is not exactly a 365 day cycle, but rather a bit more - something like 365 and 1/4. Therefore, we need to add an extra day every 4 years to catch up with the nature. This story was good enough for me but wont be for my kids. These curious monkeys will interrupt with some unexpected questions. It will be too late at night to Google it, and next day we will all forget. So, lets prepare the bases and research this a bit more.

How do we know the precise time it takes for the Earth to rotate around the Sun? Each year we have two special days: one around March 20-21st and another around September 22-23rd. We mark these days as the beginning of Spring and the beginning of Autumn in the Northern Hemisphere and on opposite in the Southern Hemisphere. At some point during these days the Sun is positioned right above the Earth's equator making day and night approximately equal in length all over the world. Remember that nights are shorter than days during winter and longer in summer. Astronomical year defines a full cycle of the Earth around the Sun and is measured in-between these "equal" Spring moments (that are called "equinox"). Right now this period is 365.242374 days and it is slowly increasing!  Increasing??? Why? Does the Earth gets older and moves slower or the Earth's orbit around the Sun expands?  Aha! Now you remember hearing that the Earth is slowly moving away from the Sun. So, that's true..

Back to the 365.242374 number. How to count the tiny remainder in our calendars?  Let's try avoiding it altogether and always count 365 days in a calendar year:  in about 4 years Spring (and all the rest of the seasons will shift by 1 day),  in 120 years by 1 month, in 720 years by 6 months and the seasons will completely swap. Imagine a heat wave on Christmas and freezing cold during a summer vacation!

So, we have to account for the 0.242374 to prevent such weather scrabble.  Every four years: 4 x 0.25 = 1 extra day. Here it is - February 29th. But wait, there is more! The remainder is not exactly a quarter, rather a bit less: 0.25-0.242374=0.007626  Small number that gains weight only when multiplied by a 100 or so. We need to weed out some leap years about every 100 years. It was agreed that years that are divisible by a 100 and not divisible by 400 will not be considered leap years (even though they are divisible y 4). For example, years 1600, 200, 2400, 2800 are leap years while 1700, 1800, 1900, 2100, 2200 are not.

Now it makes sense why it is called  a "leap" year - calendar year is catching up with the actual astronomical year to avoid snow on summer vacation. And different calendars (Julian, Gregorian, Chinese, Hebrew, Islamic etc. ) do it differently. But this is another much more complicated story.

P.S. Satisfied with the information I learned I told it to my kids at bedtime on February the 29th and my son did startle me with an unexpected question: when was the first leap year? Was it year 4?
I still couldn't find the exact answer to his question but I read that Egyptians were the first to come up with the idea of using leap day to keep the calendar in sync with the earth rotation. Julius Caesar adopted the idea in the Roman empire. However, the Julian calendar had only a simplified leap rule - leap year every four years that over the years resulted in accumulation of too many extra days. This got corrected only when Gregorian calendar was introduced in 1752 and  a leap of 11 days was necessary to catch up with the Earth and Sun.

Image by jronaldlee, distributed under CCL.

How Math Was Taught in....

A fun story arrived via email a few days ago. I am translating it from Hebrew below.

A woman described her experience in a supermarket. She was buying produce worth $158, gave to cashier two $100 bills and was looking for $8 in her wallet to avoid change in small bills and receive just a $50 bill back. Young cashier hesitantly took the $8 and was clueless why it is given to her and what to do about it. They ended up calling a supervisor who patiently explained to the cashier the math of this transaction, while she stood there with a tears in her eyes not understanding what all these people want from her.

Coming home this woman contemplated what changed have happened in math education from the 50th to nowadays that has shaped attitude of students who became immune from any criticism by the teachers. She wrote:

This is how we taught math in the Fifties:
"A farmer sold a cart of oranges for $100. The cost of production of these oranges is 4/5 of the selling price. What is the farmer's profit?"

This is how we taught math in the Seventies:
"A farmer sold a cart of oranges for $100. The cost of production of these oranges is 80% of the selling price. What is the farmer's profit?"

This is how we taught math in the Eighties:
"A farmer sold a cart of oranges for $100. The cost of production of these oranges is $80. What is the farmer's profit?"


This is how we taught math in the Nineties:
"A farmer sold a cart of oranges for $100. The cost of production of these oranges is $80. What is the farmer's profit?"
Mark the correct answer:
() $20   () $40  () $60  () $80  () $100

This is how we taught math in the year 2000:
"A farmer sold a cart of oranges for $100. The cost of production of these oranges is $80. Farmer's profit is $20. Mark below if the last statement is:
() True    () False

This how we teach math now:
"A farmer sold a cart of oranges for $100. The cost of production of these oranges is $80. If you know how to read mark X next to the number $20 because this is the farmer's profit..."
() $20 () $40 () $60 () $80 () $100

Playful Math in Advertising

 Ad agencies are usually avoiding any serious math. However, some campaigns demonstrated that simple math can intrigue, stick in memory and sell. I decided to start a math ad collection. If you recollect any relevant ads - please share them.

The 1998 US Godzilla movie ad campaign has been gripping LA for months. I remember a giant lizard foot covering a whole city bus and words: "Size does matter."

Sheraton Hotels came with this surprising promotion in the September of 2011.  Youngsters win, older people can sue for age discrimination.

In 1999 Lord and Taylor department store's holiday decorated window claimed: "Great dress is worth its weight in gold or silver." And the beautiful, T-shirt style, holiday dress was there, sparkling behind the ad. The dress came in gold and silver colors, was sold for $108 and probably weighted 108 grams.

A provocative Zip Car ad:

A strange Nova Scotia toys company ad (don't try to solve it):

Myths about Gender and Mathematics Performance

True or False:

1. Girls under-perform boys in math.
2. Boys are born with larger innate intellectual potential and this explains why boys are and have always been out-performing girls in math.
3. Even if boys and girls math performance may be similar on average, there are many more boys with extremely high math capabilities than girls.
4. Women’s nature include a tendency to prefer the more nurturing fields, such as nursing and teaching young children, to the more quantitative ones, such as mathematics, physics, and engineering. Therefore it may be a waste of time and money to expend resources directed toward trying to increase participation of women in these mathematics-intensive fields.
5. Separate education of girls and boys in the single-gender schools improves math performance of girls. 

As you may have guessed neither of the above statements is true. And if you doubt it, take a look at the recently published scientific article by Jonathan M. Kane and Janet E. Mertz "Debunking Myths about Gender and Mathematics Performance"

My short summary of the article's highlights: 

The study above compared performance of boys and girls  from 65 countries  in a variety of tests and found no statistically significant difference overall in boys-vs-girls performance. In some countries, like Bahrain, girls slightly outperformed boys in math, while in others, like Tunisia, boys slightly outperformed girls. Interestingly, it has not always been this way. In the US a significant gap existed between the performance of boys and girls in the 1970th. The percentage of Ph.D.’s in the mathematical sciences awarded to U.S. citizens who are women has increased from 6 percent in the 1960s to 30 percent in the past decade. Ratio of boys-to-girls among those scoring above 700 in the SAT has changed from 13-1 to 3-1 between 60th and 90th.  This suggests that sociocultural, legal, and educational change that happened in our society between 60th and nowadays is responsible for the advancement of girls in math. 

OK, girls on average are performing as well as boys in math. What about the fact that there are many more boys at the top level of math elite? Is this nature or nurture? The authors compared number of girls who excel in mathematics performance at an extremely high level among many different countries and concluded that that it very much depends on the equality between men and women with respect to economic participation, educational attainment, political empowerment, and health.

How about the single-gender schools? The study analyzed test results of 8th graders in the 17 countries. Indeed, math advancement of girls in some countries may be explained by the single gender schools. However, the study concludes that this is not a rule and most likely caused by under-performance of boys in these countries or different educational experiences and patterns of school attendance.

In general authors write that mathematics performance of students largely reflects the academic standards and expectations of the community in which they are raised.  Specifically, home environment is a primary determinant for success of children in school.

Top image by Demi-Brooke, distributed under CCL.

Two New Math Apps

Both of these apps will help your kids master addition, subtraction, multiplication and division. They do not teach but provide fun rigorous practice for elementary school kids.

Bubbling Math for iPad and Mac by Tappy Taps.

This type of graphics will probably be most attractive to the girls (age 5-9) . The interface is very intuitive for novel as well as experienced players. Parents or kids can select level of complexity for each kind of the operation. Parents can review the day-to-day progress and wrong answers. As child progresses new lands and places are unlocked. I wish there would be more to the story that would push your child keep going or alter the plot based on the child's responses.  Still, it is a great beautiful alternative to simple math sheets. iPad app is $1.99, Mac version is $3.99 but now if FREE.

Operation Math Pocket by Spinlight.

Terrific comics-style graphics for older kids (9-12), especially boys. Kids need to be comfortable with reading and comprehension to understand the overall plot. I wish there would be a woman agent to attract girls as well. As you solve +,-,*,/ exercises you unlock some spy gear and accomplish spy missions all around the world. Nice, fast-paced, engaging. As with the Bubble Math app, it could be great to get to the next level of interactivity where the story is feasibly affected by child's responses. However for the price of $2.99 on iPad and $1.99 for iPhone/iPod (during the limited offer-launch sale) this beauty is a bargain.

Can you trust the mirrors?

If one of your New Year's resolutions was to shape up, I have good news for you: You look much thinner than what you think you are. Don't believe me? Take an old lipstick or washable markers, stand in front of a home mirror and carefully draw your outline on a mirror. Now measure the size of the face you just drew. It would be about half of what you expected. Viva-la-mirror!

How can it be that you on a surface of a mirror is about half the size of the real you? Here is the math:

Take a look at the triangles ABC and DEC.
Your face and its virtual image in the mirror have a size AB. The projection of your face on the mirror surface is the red line DE. AB is twice the size of DE because CH is twice the size from you to the mirror.

Are you really half the size? No, I was just kidding.
Our brain focuses on what is reflected in the mirror, rather than on the surface of the mirror. AB and not DE.

This peculiar trick is from a great book by John D. Barrow "One Hundred Essential Things You Didn't Know You Didn't Know. Math Explains Your World."

Top image by Dru!, distributed under CCL.

Baby Feeding Math

We have a new baby in the house and with her arrival came various logical riddles that appear very simple, yet the solutions are not trivial although can mean quiet, health and happiness of the whole household.

What time did I last feed her?
What breast did I last feed her from?
How much did she drink?

While I have been trying to remember the last feeding times, I have not always been able to keep track of the "last breast." In many cases you can feel the pressure of the milk, however not always - just recollect the sleepless zombie state new parents find themselves in. I mentioned this problem in my newsletter and many of you wrote suggesting placing a sticker on a bra, writing L or R on a notepad, moving a ring from one hand to another. Someone proposed this as a great idea for a startup.

Today another creative solution arrived by mail from the Similac baby formula company. A present or an advertising trick, depending on how you look at it.  An elegant hand bracelet:

Instructions advise to put the bracelet on the hand corresponding to the side of the last feeding (when breastfeeding) and slide the heart-shaped window to the mark corresponding to the time of the last feeding. Two riddles are solved with one bracelet! Very neat - I am already using it.

Another puzzling and usually annoying question that is frequently asked by nurses, doctors or grandmothers is: "How much milk does your baby consume in one feeding?" When baby is on a formula - it is easy to answer. When breastfeeding - the only way to answer this would be to pump, measure, then feed.  Too much work just to satisfy the curiosity. Feeding time may give a clue to the answer, but as experience shows milk sucking speed depends on the baby's age, temper and hunger state. What other optios are there? You may weigh the baby before and after the feeding or weigh the mom, but the 100-200mg consumed will likely get lost in the noise of the weighting scale or a burp.

Undoubtedly various creative solutions to these questions are in the works.  Have your own ideas - rush to patent it; use someone else's idea - share it with us.

Multiplication Song

Like many of you, I have memorized the order of the letters in the English alphabet thanks to the "ABC" song. In my case it happened at the age of 30, from my kids' Sesame street DVD (with Billy Joel). Yes, I learned the alphabet before in school but it never settled so naturally inside my brain until I heard it fused with music (and over and over again). No matter that I am practically tone deaf.

My daughter quickly mastered the Hebrew alphabet also from  a song - "Alef, Beth" by Naomi Shemer.
My late grandma surprised us all starting to sing some childhood songs in Yiddish at the late stages of her Alzheimer, when she no longer recognized any of us.  This all is not a coincidence.We are more likely to remember a poem that was put to music than "bare" rhymes.  Scientists have shown that music facilitates quick memorization, long-term memory retention and efficient retrieval of information. So, shouldn't we be singing more at school and work?

In the past few months, a large amount of readers and friends consulted with me on how to help their 3rd graders memorize the multiplication table. Well, I have been struggling with this with my own daughter and nothing seems to do the magic. We tried memorizing the diagonal, or starting from the upper-left corner and slowly increasing the square size, we practiced on windows of buildings that we passed by, we tried online games and iPhone games. She got better and better at it but was easily forgetting complex parts after a few days without practice. Plus, it was not fun at all.  Perhaps we should put the multiplication table to music.

I checked on YouTube and found a few attempts by various artists. If you are aware and like some other clips - please add the link below or email me.
Share this with your kids. Better yet, write your own!  There are few professional musicians among our readers and I challenge you all to try - contribute a rhyme, or music or both.

The Multiplying Multiplication Song !!!


Mrs. D's Multiplication Rap Remix™

How to make a festive paper decoration

My daughter convinced me to share with you the tips and tricks on making this wonderful paper decoration for Christmas,  Sukkot, birthday or any other celebration.

She made it in school and I noticed how every math-curious person who came into our house in the past week (including my father and cleaning lady) carefully folded it and tried to reverse-engineer the creation process. We did it as well and after a few failed attempts finally managed to figure it out.


You will need: two pieces of regular (A4 size, white or better colorful) paper, scissors, ruler, pen, eraser and stapler.

Start with a full sheet of paper.

fold it in half and again in half:

Hold on from the folded corner and cut the open corners of the sheet diagonally:

You are left with a folded piece in the shape of a triangle. Use ruler and pen to make the following lines on this triangle.  Note that lines are parallel to the triangle's base. They start at the opposite sides of the triangle and neither line goes all the way to the other side:

Cut along the lines:

Unfold the triangle:

and unfold again very carefully:

Gently tilt the middle part so that you can hang the decoration from it:

You got it!

Now you can make another one of those and staple them together:

Enjoy:

And now when you mastered this try reverse-engineering this Escher's drawing:

Online Math Games: Bad or Good for your kids?

You get a link from your child's teacher. You load the page and easily interest your 8-year-old in the selection of colorful and innocent math games. Alien addition, sailboat subtraction, penguin multiplication, drag race division and tens of others. You point her to multiplication that you both have spent the entire August practicing. You patiently wait while she spends 5 minutes on choosing the color for her penguin and then leave her in front of the naive penguins jumping from iceberg to iceberg. You go to fill the dryer with laundry or check your email with a cup of coffee. When you return, you are most likely to find your darling in front of youtube, pbskids, frustrated and certain that she is bad at math. How did this good turned bad so quickly?

I have always been ambivalent about recommending online math games. CD-ROMs - yes. We have a few excellent ones in my household and both kids enjoy playing them for hours. They are grade-based, start you on the Easy level, and do not rush or humiliate you. With online games it is a different story. Many of them, such as recommended at my kids' school Academic Skill Builder Games take into account the speed of answering along with the correctness of results. Every game is a race where for starters you are guaranteed to come last unless you are choosing a game way below your skill level.
Ready, set, go.
The timer is rushing you.
You notice the red, blue and green penguins are getting ahead while your pink cutie is sinking in the water each time you confuse 6 x 1 and 6 + 1. No time to recover. You press the wrong button again, you squeak in frustration, you lick your tear. Game is over. Your name is last in the results list alongside the naive little pink penguin. Math is fun, isn't it?

A perfect five minute scenario for a quick bloom of serious math-phobia. Beware of online math games. Never ever let your kids start them alone. Instead - demonstrate and test the game on yourself first. Let kids see you struggle with the keyboard, the rules of the game, the penguin sinking disaster, let them see you lose and not give up. It may take a few trials. When you finally manage to finish first, play it together with your kid with one of you answering and another pressing the buttons. Finished in the first three? This is fun. Now you both are ready for the full transfer of the game power to the kid. The level is right, the rules are clear, kid has warmed up and ready for an exciting math practice.

If you are away at work and have to instruct your kid to play such games remotely - recommend to them to start on a very easy level way below their grade skills, carefully reading the instructions and slowly raising higher as they get more comfortable and confident with the speed and rules of the game.

These are my thoughts. Teachers and parents - what is your experience, strategies, trials, errors and successes in the online math gaming world?

It depends on how you measure it

I always thought that cooking is simple as long as you have a reliable recipe. Over the years I learned to trust some of the newspaper food columnists, while taking with a grain of salt my kids’ class cookbooks because some words fade during the kids’ mail transit and Xeroxing.

Once you have a good recipe there is usually no one to blame for a bad outcome other than yourself. Well, occasionally you can tell that the eggs were too big or the stove is new. My eccentric grandaunt used to prohibit us from entering the kitchen during baking and blame us for disturbing the cake’s gestation period if it didn’t rise. But there is really not much more. One cup of flour is one cup of flour anywhere in the world, one tablespoon of butter is one tablespoon of butter. Or is it?

This week’s article in the New York Times surprises us by suggesting that it is not. It refers to an experiment when ten different people were asked to scoop 1 cup of flour and pour it into a bowl. The weight of flour in the individual bowls varied between 4 and 6 ounces depending on the strength and technique of scooping used by each participant. This meant that some of these people may be making a cake with 1.5 times as much flour as others.

What else can we use instead of the traditional and universal cup and spoon volume measures? The weight (mass), says the article advocating for simple kitchen scales. Note that weight is equal mass as long as we cook on earth.

Image by jamieanne, distributed under CCL.

Let’s recollect some math and physics:

Mass = Volume x Density

If the mass of 1 cup of flour in the bowls varied from 4 to 6 ounces, it means that the density is to blame. Faster scooping, scooping up vs down techniques, different storage, type of flour, shape of the cup – all of them can influence the density of the flour. To get the same amount in your recipe rely on mass.


The difference may be even more drastic when dealing with grated cheese. According to the article “the heavier shavings of a box grater can fill a cup with twice as much cheese as” “billowy ribbons of machine-shaved cheese.”


So, get yourself a kitchen scale for the next holidays. Use mass-based recipe source and you will:

1. Get consistent recipe-matching results every time.
2. Easily double or halve the recipe.
3. Have less stuff to clean. You can use only one mixing bowl by slowly adding ingredients into it directly from the containers and zeroing weight on the scale after each addition.

What about your old favorite volume-based recipes? One cup of oil in mom's sweet corn bread, one cup of honey in the Rosh haShana cake. Should you just convert them to mass?  This Pyrex measuring cup clearly marks 1 cup volume as 8 oz mass.

Remember the formula:

Mass = Volume x Density

For water measurements: 8 oz = 1 cup x Water Density
Oil, melted chocolate and honey are obviously denser than water. Higher density gives higher mass: around 10 oz for one cup of oil, and 12 oz for one cup of honey. So, beware of the Pyrex' cup.


It is just you now in the kitchen with your scale and math.

Amazon links to buy kitchen scale:

Presenting Math as an Art

I was disconnected from the internet for most of the August and got a luxury of time to read a few deep-hidden non-fiction books from our library. One of these books was a shocker - a light, eloquent, captivating and constructive critique of the ways mathematics is percieved and presented at schools, universities and most of our households. It is a very refreshing and convincing analysis from a former scientist and experienced math teacher and I am going to recommend it to every teacher, parent and high school student. The book is: "A Mathematical Lament" by Paul Lockhart. It is subtitled: "How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form."

Dr Lockhart suggests that math is an art and should be treated exactly like music and painting. We should not be learning math because it is applicable to our daily lives or essential in our future occupations, but rather because the process and the result bring joy to our soul like listening to or producing music does. Forget all the bad math associations you have nurtured and think of a pleasure that discovery of any pattern brings to us. And the sweet mental game of attempting to explain this pattern.

Take for example this curiosity:

1 = 1

1+3 = 4

1+3+5=9

1+3+5+7=16

1+3+5+7+9=25

Fascinating! Sum of the subsequent odd numbers produce results that are all square numbers.

But why?

Can any square number be represented as a sum of subsequent odd numbers?

If we let our mind wonder about this for a while we may be able to come with an explanation, perhaps even as thrilling as this one:

Any square number can be drawn as a square, such as this big and colorful 5x5. Any square can be split into such colorful parts.

Count the number of little pieces in these parts. They are 1 (purple), 3(green), 5(yellow), 7(blue) etc. These are our subsequent odd numbers. Looks like any square can be split into such pattern.

Lockhart writes that in school "the rich and fascinating adventure of the imagination has been reduced to a sterile set of facts to be memorized and procedures to be followed." Being math expert, math lover and experienced math teacher he suggests a number of directions to improve such math education:

• "Mathematics is an art of explanation." This means that students should be allowed to pose their own problems, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble togather their own explanations and proofs."
  
• Math history is fascinating and provides a great base for captivating storytelling. Instead of handing kids formulas to memorize retell them stories of ancient and contemporary mathematicians and their search for solutions to the fascinating problems that are wrapped in these formulas today.
• Like other arts math should be subject to debate and critisize in schools. "Is this argument sound? Does it make sense? Is it simple and elegant?"
  
  
• Forget proper notations that are frequently a source of frustration. "Math is not a language, it is an adventure." DaVinci, Pollock,and Warhol each created art in their own way.

The author also cleverly warns that although math is and should be presented as an "art" subject, such shelving may be dangerous as "useless" art disciplines are frequently the first to be cut off from the curriculum in time of budget cuts.

This book will definitely change the way you think about math.

It may boil your blood from anger with the author or on opposite, the current math curriculum against which he convincingly stands.

It may explain you why you always hated math classes but liked riddles and puzzles.

It surely will inspire you and teach you a few tricks that you will rush to share with your kids as I did.

Daily Roadside Math

Walking around Vancouver today I noticed this stretched bicycle sign:



Why is it so distorted? Was it painted by a machine that accidentally accelerated shortening the bicycle at the top?

There must be a reason for this distortion. Probably something facilitating our perception. Similarly to the Ambulance sign painted with a horizontal flip on the ambulance truck to allow the sign be correctly read in a rear view mirror.

Let's see. Who is it for? A warning for approaching cars and bicycles that this road is bicycle only. They should be able to see it from 20-50 feet away. Let's check how the sign looks from such distance:

Just perfect!
What exactly is happening with this visual illusion?
The drawing of the bicycle is cleverly made to account for the perspective projection distortion in our eyes (and brain) and allows for the bicycle to be perceived clearly from the average height and far distance. Note how the bicycle wheels appear perfectly round on the second picture.

This is simplified demonstration of our eye perceiving bicycle wheel as perfectly round from a distance and as an oval from close up.

Now, how about an explanation to this signage?

From Magic Squares to Sudoku

In 1415, the German renaissance painter and mathematician Albrecht Dürer created this busy copperplate engraving of a woman melancholically contemplating amongst a clutter that includes an hour glass, a balance scale, a sphere, a polyhedron (aka philosopher’s stone), a purse, and keys. Like his contemporary’s Mona Lisa, this mysterious painting has been extensively studied, speculated upon and written about. It is believed that it was planned to be the first in a series of paintings describing various temperaments: melancholic, phlegmatic, choleric, and sanguine. It is titled: “Melencolia I”




Among the peculiar geometrical tool in this painting, in the upper right corner, one can find … a magic square!



Wait; weren’t magic squares invented by Ben Franklin three centuries after?

Magic squares are tables of numbers where sums of every row, every column and both diagonals produce the same number. In Drurer’s square the sum is 34:








Magic squares have been around for thousands of years. Magic squares of size 3x3 are mentioned in the Chinese literature dating 650 BC. Magic squares of size 5x5 and 6x6 are found in an encyclopedia from Baghdad written in the ten’s century. They were engraved in stone and metal, worn as talismans, their usage believed to ensure longevity and prevention of diseases (from Wikipedia). Magic squares can be spotted in art all around the globe. The Indian Parshva temple still contains a 4x4 magic square carved in the 10th century. The Passion facade of Gaudi’s Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4 magic square with a sum of 33 that is the age of Jesus at the time of Passion.



What has fascinated people about Dürer’s square is that in addition to the regular magic properties, it contained more mathematical patterns and messages:

1) Numbers in each of the following five quadrants add up to 34.



2) Any pair of numbers symmetrically placed about the center of the square (such as these) sums to 17:



3) The two numbers in the middle of the bottom row give the date of the engraving: 1514



4) The numbers 1 and 4 at either side of the date correspond to, in English, the letters 'A' and 'D' which are the initials of the artist.

What about Ben Franklin? As a child and an adult he amused himself with creating large size magic squares such as this 8x8:



While his squares are not truly magic squares because sum of each of the diagonals is not equal to the sum of numbers in every row or column, they are still pretty cool. To compensate for a lack of the diagonal property Franklin defined a broken diagonal or “bended rows” properties where the sum of numbers in each highlighted bended rows is the same.



Magic squares could be considered the ancient relatives of the modern Sudoku game. In the late 19th century, French newspapers started publishing matrix puzzles – 9x9 magic squares with some numbers removed. Then the puzzle slowly evolved taking away the sum requirements and instead requiring that each row, column and diagonal have all the numbers from 1 to 9. And finally, the 3x3 sub-division was added producing the modern day Sudoku (you can click to play):



Thousands of Sudoku are published in the world daily. You have likely seen someone doing Sudoku on his way to work, at the playground, laundromat or in the coffee shop. A good story attesting to Sudoku’s popularity was published in the Sydney Morning Herald. In June 2008 an Australian drug-related jury trial costing over Australian $1,000,000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to the evidence.

When to expect when you are expecting

They were expecting their third child – a daughter – a week after Labor Day. Perfect timing: their older kids will be out of the house and back to school, and the doctors back in town. All was well except that a few months before he realized that the September 12th due date fell inconveniently close to a professional conference in Europe, which he organized. “To go or not to go?” was pulsating daily in his brain. Will she manage here with 2.5 kids on her 9th month of pregnancy without me? What are her chances of delivering 3 weeks earlier? Shall I bring my mother to help here instead of me? If I stay – something will surely be messed up at the conference. Who will do my part? What would my colleagues say about me abandoning them? It is an honorable responsibility and I want to see it through.

***

Due date is calculated based on the empirical observations that it takes an average of 38 weeks from the moment of conception to the moment of delivery. But no one knows when exactly conception took place and how fast the sperm moved and fertilized the egg, plus it may be embarrassing to admit that you did it while at your in-laws or on a friend’s couch. So we look around for the closest related landmark – the menstrual period. Its first day usually is 10-18 days before the conception. So, your due date is approximately:
38 weeks + [10-18] before conception days =roughly 40 weeks from the first day of your last menstrual period
And ironically enough, when you conceive you are already considered 2 weeks pregnant.

The actual birth dates are spread around this 40 week due date in approximately normal distribution (bell curve).

(Graph from Ref 1.)
Graph mark of 24% at the 40th week mean that your chances of delivering on any day of this week are approximately 24%/7=3.4%. You can see that around 90% of babies are born within three weeks of their due date, and 21% are born within 3 days of it. In many geographical places, the right side of this distribution is sharply raised and cut at 42 weeks because the majority of the women that have not delivered by the 42nd week are being induced.

Why does the gestational age vary? Because not everyone ovulates and conceives exactly 14 days after the beginning of the menstrual period; some babies mature faster while some are too comfy inside; second and further babies seems to be more in a rush to come out; mother’s age and weather may influence the dates; and women of some ethnicities tend to give birth a bit earlier than others. (Ref 2, 3, 4)

Back to our expecting couple. He took from Wikipedia the data about probability distribution of actual births with respect to their due date.



He centered the graph on their due date, September 12th. Total area under this bell curve is equal to 1 = accumulated probability of giving birth. The chances of delivery during any interval are equal to the area under this interval. He marked in red the probability of her delivering prior to August 27th while he is away. He added up all the red area under the curve obtaining 0.043. He proudly showed it to her, explaining that her chances of giving birth before he returns are only 4.3%.
She told that this may be true but he is a 100% asshole. She also reminded him that both of their older kids were born two weeks earlier.
He still went abroad and run the conference.
She gave birth a week and a half after his return (on September 8th) to a healthy and beautiful baby girl.
They lived happily ever after, but he was never forgiven.


References:

1. Calculating Due Dates and the Impact of Mistaken Estimates of Gestational Age
2. Gestational Age at Birth, 1994 and 2004
3. American Pregnancy Association
4. Duration of human singleton pregnancy. A population-based study.

iPhone and iPad apps of TheMathMom Stories and Puzzles

New: iPhone and iPad apps of TheMathMom Stories and Puzzles. Get them while they are free. Hundreds have been downloaded in the last week since these apps were released. Read, answer or share from anywhere on the go. Click on the icons to download apps from iTunes. Please share your feedback on iTunes.

Spring IQ Test

Ready to shake up your brain after the long and cold winter?
This IQ Test is easy, fun and playful. No math skills above Kindergarten level are required. Answer quickly, without hesitation.

Puzzle #1: You participate in a race and you are advancing ahead of a racer who is second. What is your place in the race right now?

Answer #1: If you answered that you are on the first place - you are absolutely wrong. If you went ahead of the runner who was second, you now took his second place.


Puzzle #2: Two fireman entered a school and one of them claimed to be a father of the other's 2nd grader. Can this be true?

Answer #2: Hopefully, you have answered yes. No complex family relationships are necessary, the other fireman must have been the 2nd grader's mother.

Puzzle #3: A girl entering a library was surprised to see her friend walking toward her. The friend was accompanied by her mother, two grandparents, younger brother and a little puppy. The friend was also holding a doll. How many people in total just came to the library?

Answer #3: Don't even think of answering 5 or 6. They are all leaving the library. Only one girl just came in.

Puzzle #4: How many pairs of animals did Moses take on the Arc?

Answer #4: It was Noah, not Moses.

Puzzle #5: Mary's father has five daughters: 1-Arlene, 2-Darlene, 3-Marlene, 4-Sharlene. What is the name of his fifth daughter?

Answer #5: If you answered anything ending with "lene" - wrong again. Run this puzzle by your toddler kid. She will help you figure out that Mary's father should have at least one daughter named Mary.

Better luck next time. Enjoy sharing this.

Top image of the brain-resembling coral is by boogieswithfish, distributed under CCL.