Perfect Practice Makes Perfect

# Articles

## Happy Square Root Day!

April 4th, 2016 by John Lehet

“Square Root Day? What’s that? That was my response last week when a student in my Calc class mentioned that Monday 4/4/16 was Square Root Day. Up until that time, I had never heard of Square Root Day. My student proceeded to inform me that Square Root Day’s occur when both the day of the month and the month of the year are square roots of the last two digits of the year! For example, April 4th, 2016 is day 4 of month 4 of year 16 of the current century (zero based for those of you counting).

Unlike Pi Day, which happens yearly, Square Root Days occur much less frequently, happening only nines times each century. The last Square Root Day was 3/3/2009 (that’s 7 years, 1 month and 1 day ago) and the next Square Root Day will be 5/5/2025 (that’s 9 years, 1 month and 1 day from today). Therefore, in a given century, Square Root Days occur at increasing periods of time, each of which is an increasing, consecutive odd number of years away.

This Square Root Day is an interesting one because it is the middle of a Pythagorean Triple – 3,4,5. Consecutive Perfect Squares (of integers) differ by an odd number (just like the years between Square Root Days in a given century). That odd number can be further quantified as 2n+1 where n is the square root of the perfect square. For example, given the perfect square 16, n is 4 (the square root of 16) and 2n+1 is 9 which is the distance to the next perfect (i.e. 16 + 9 = 25, the next perfect square). So whenever 2n+1 is a perfect square, a Pythagorean Triple occurs! Therefore, every odd perfect square greater than 1 (e.g. 9, 25, 49, 81, 121, 169 …) is part of a Pythagorean Triple.

Let’s see if this works for 23.
– 232 = 529.
– from 529, Subtract 1 then divide by 2 to get 264.
– add 1 to this to get 265
– there’s our Pythagorean Triple, 23, 264, 265.

Check it out – 232 + 2642 = 529 + 69696 = 70225 = 2652.  There we have it, not only is it Square Root Day, but we have a way to generate a Pythagorean Triple using any odd number (greater than 1)!

If by chance you have a newborn today, may I suggest the names GiGi, MiMi or Lulu for a girl or JoJo for a boy, all of which are “perfect squares” (of a sort). Have a great Square Root Day!

## Happy PI Day!

March 14th, 2016 by John Lehet

It’s that time of year again – March 14th – that’s right, it’s PI DAY! PI is an irrational number. No, that doesn’t mean it’s unreasonable.  It means it’s not rational – that is PI can not be represented as a ratio of two integers. In other words, PI can not be represented as a fraction (where the numerator and denominator are integers). However, PI can be approximated as a fraction or a decimal number. Many know PI as 3.14 or 3.1415 or even 3.141592653 (knowing this many digits promotes one to dork status – but that’s alright by me). No matter how many digits, these are just approximates of PI, often well within given or acceptable tolerances. Computers have found decimal approximations of PI to over a trillion decimal places (I kid you not!). There are many very interesting properties associated with PI. One that I find the most intriguing is the following summation:
So multiplying the right side of the equation by 4 will result in an approximation of PI.  Although PI can not be represented as a fraction, it can be found by infinitely adding and subtracting fractions (this is called an infinite summation).  Even index terms are added and odd index terms are subtracted.  Each term is just the reciprocal of 2n-1 (e.g. 1/(2n-1)) where n is any natural number.  Initially, the summation does not show much promise.  The sum of the first 12 terms is approximately 2.8952 – not very close to PI.  However the sum of the first 200 terms is approximately 3.12159 – much closer!  I have created a quick program that computes the approximate of PI based on the number of summations you input.  Try 1000, 100000, 1000000 (a million) and 1000000000 (that’s a billion – it takes a few seconds) to see how the approximation converges towards PI.

Have fun with PI and HAPPY PI DAY!

## Up A Tree

October 25th, 2015 by John Lehet

Life must be lived as play.” – Plato

“What’s that?” she said looking out the window.

“What’s what?” replied her husband, his eyes remaining glued to the morning paper.

“That!” she replied emphatically, pointing out the window.  “There, out the window.  Look why don’t you?”

Her husband had no choice but to put the paper down.  He squinted and glanced out the window.  “It looks like a boy to me.”  He said, slowly raising the paper and back to the all important news.

“Of course it’s a boy.  I know that!  I’m not stupid you know, but what’s he doing”

Without lowering the paper or even taking his eyes from it, the man replied “It looks to me like he’s climbing a tree.”

“Of course he’s climbing a tree, but why?  Doesn’t he know it’s dangerous?  I can barely look myself.” She said, never taking her eyes from the boy.  “He could fall and get hurt, maybe even kill himself.  Now he’s stopped.  I wonder what he’s doing?”

To this query, her husband made no reply.  A short moment of silence ensued while he maintained his news gathering vigil.  His wife kept both eyes focused on the boy and then suddenly she blurted “There he goes!  He’s going up again!”

Without moving his eyes from the paper, the man replied “Of course he is, it wouldn’t be climbing otherwise.”

“Very funny!  I don’t find any of this funny.  I think it’s dangerous.  I think it’s crazy.  It should be against the law!  I’m calling the police!”

“The police?  What for”

“The boy, he could get hurt.  I wonder where his parents are.  I bet they don’t even have half a clue that he’s out there climbing trees.  Climbing trees, it’s all crazy to me.  Why in the world would he want to do that?  He must be on drugs.  That’s it, he’s on drugs.  It has to be drugs.  He must be hallucinating or something like that.  I’m calling the police before he hurts himself, or worse yet he hurts someone else”

“Hurt someone?  What’s he going to do fall on somebody?”

“Oh, very funny, Mr. Funnypants!  You won’t be laughing when he breaks into the house and shoots us!  That’s what people on drugs do, you know!  They have no idea what they’re doing, they’re totally out of control.”  With this, she started to rise to make her way to the phone, but suddenly stopped.  “Oh my god!”

Even her husband took note of her excited tone and looked up from his paper.  She continued “Now he’s just hanging from a branch!  What in god’s name is he doing that for?  He’s at least fifteen feet from the ground!  I can’t watch.”  By this time, her husband was standing at the window, getting a better look at the boy.

Now, he too was concerned.  “Forget the police, call the fire department!  He may fall.”

His wife jumped to the phone and dialed.   When connected she hurriedly said “There’s a boy climbing a tree near our house.  I think he’s stuck. . . No, not a cat, a boy. . . I think he’s on some sort of drugs. . . No, he’s not playing a computer game, he’s really up a tree . . . No, I haven’t been drinking, but he may have been . . . How should I know why he’s climbing a tree, he just is . . . I personally think he’s on LSD or something like that . . .Well, help him get down of course!  That’s what you do isn’t it?  . . . Fire, no there’s no fire, but if he’s on drugs, he may start one . . .  Insurance?  How should I know if he has insurance? . . . you can’t!  . . . Police jurisdiction . . . OK, I’ll call them.”

The woman hung up the phone and said to her husband “They said to call the police, it’s their jurisdiction.  He didn’t think that it was against the law to climb trees, as long as he’s properly insured.  It must be against the law.  Who in their right mind would climb a tree anyway?”

“Maybe he was being chased.  If I was being chased by a dog, I might climb a tree.  That shouldn’t be against the law.”

“Do you see any dogs?  Did you see anyone chasing him?  Use your head for once.  He was hanging from a branch!  Would you hang from a branch, if you were chased by a dog?”

With reasoning and experience, the man saw the error of his ways.   He had just recalled times in his youth when he climbed trees.  He climbed them for fun, for adventure, for something to do.  He would hang from branches in a minor attempt to challenge fate, knowing fully that he was in control.  Knowing that he was in no real danger.  He remembered his youth and how much fun it was.  He innocently said to his wife “Maybe he’s climbing the tree for fun.”  Instantly, he again saw the error of his ways.

“Fun?  You call that fun?  Endangering your life and possibly other’s lives!  Making you mother worry for no reason!  Fun?  If he wants to have fun, he should be inside safely playing a computer game.  Doesn’t Wii make a tree climbing simulation?  I hear it’s just like the real thing, even better some say.  Fun, my eye!  I think you’re on drugs, that’s what I think!”

The man just glanced out the window, enviously recalling his youth.  His wife called the police.  They said they’d be right over.  They’d surely rid society of this menace, she thought.  But before they arrived, the boy climbed down the tree and innocently trotted off across the field and out of sight.  Moments later, the police came.  The woman showed them the tree, pointing up, showing where the boy had been.  The police shook their heads in disbelief.  They then searched for any evidence of foul play, but found nothing out of the ordinary.  The police left the scene and the woman made her way back to the house.  As she neared the house, she was met by her husband.  He walked past his wife towards the tree.  She stopped, stared and slowly started to follow him in silence.  He came to the tree and looked up it.  His wife, lagging a few paces behind, also came to its base. She asked, “What are you doin’?”

The man lifted his right foot and without looking at his wife replied “I thought I saw a dog” and he started to climb.

## Tuesday’s Twister #9 – Topsys & Turvys

March 10th, 2015 by John Lehet

“Happiness is a how; not a what. A talent, not an obect.” – Hermann Hesse

Years ago I discovered one of the most interesting picture books I’ve even seen.  It’s Topsys & Turvys by Peter Newell, first published in 1902.  A Topsy & Turvy is a picture (drawings in the book) that shows one scene when viewed right side up but shows another scene when viewed up-side down.  In the book, Newell annotates each drawing with a creative couplet, one line for each view.  In all, Newell’s book contains 74 Topsy & Turvy pictures.

Intrigued by the book, its pictures and Topsy & Turvy pictures in general, I set off on goal of creating my own Topsy & Turvy picture.  I quickly found this task to be very challenging, but I persevered and created a few Topsy & Turvy pictures of my own.  Recently, I animated some of my original drawings (with Adobe Flash) so that they will turn upside-down on request.  I have included two interactive Topsy & Turvy puzzles below.

For each Topsy & Turvy picture, just select “click here” to turn the picture.  The first is called “Little Johnnie”.

The second Topsy & Turvy picture is called “Tricky Nik”.

I hope you enjoyed these original Topsy & Turvy pictures.  Challenge yourself and others to come up with an original Topsy & Turvy, it’s fun and rewarding.  I will preset more in future posts.  Please pass these onto others who may enjoy them and remember, please click LIKE below if you like them!

In order to use the interactive portion above, you need to have Adobe Flash enabled.

## Happy Birthday – Lewis Carroll

January 27th, 2015 by John Lehet

She generally gave herself very good advice, (though she very seldom followed it).” – Lewis Carroll

183 years ago, on January 27th, 1832, Lewis Carroll was born.  Lewis Carroll was just a pen name.  His actual name was Charles Dodgson.   Although he is best known for two books – “Alice’s Adventures in Wonderland” and “Through the Looking-Glass“, he was actually a mathematician at Oxford.  In addition to his writings and math, he loved nonsense.  Nonsense in the form of puzzles, riddles and poems, all of which he loved to develop and pass along to others.

One of his favorite type of puzzles was called a Doublet.  In a Doublet, you start with a word and try to form a second word by changing one letter at a time.  Each time a letter is changed, the new word must be a real word (and not just gibberish).  Here’s an example that changes FLOUR to BREAD

FLOUR
FLOOR
FLOOD
BLOOD
BROOD

Each new word has just one different letter than the previous word in the list.  Here are four puzzles for you to try in celebration of Lewis Carrol’s birthday.

1. change SHIP to DOCK
2. change OPEN to GATE
3. change CRY to OUT
4. change LIES to TRUE

These four puzzles are difficult and not necessarily for younger puzzlers.  I hope you enjoy this post. Please pass this onto others who may enjoy it!    Also, click “Like” below if you like it!

## Happy Birthday – Ben Franklin

January 17th, 2015 by John Lehet

“Tell me and I forget. Teach me and I remember. Involve me and I learn.”– Ben Franklin

309 years ago, on January 17th, 1706, Ben Franklin was born.  Franklin was indeed an amazing individual. He is representative of individual accomplishment and a passion for life.  He was the first and best known early-American Renaissance Man.   He was an author, scientist, inventor, businessmen, printer, statesmen and diplomat (just to name a few).  He also loved math, especially recreational math.  He developed many original and unique magic squares.

Franklin published a yearly almanac, Poor Richards Almanack.  This contained, among other things, many original quotes and sage advice.  Today I would like to celebrate Franklin’s birthday by sharing some of my favorite Franklin quotations.  There is no specific theme, just ones that I really like.

1. “An investment in knowledge pays the best interest.”
2. “Either write something worth reading or do something worth writing.”
3. “Lost time is never found again.”
4. “Early to bed and early to rise makes a man healthy, wealthy and wise.”
5. “Energy and persistence conquer all things.”
6. “We are born ignorant, but we must work hard to remain stupid.”
7. “Any fool can criticize, condemn and complain, and most fools do.”
8. “Well done is better than well said.”
9. “Diligence is the mother of good luck.”
10. “He that is good for making excuses is seldom good for anything else.”
11. “Being ignorant is not so much a shame, as being unwillng to learn.”
12. “Never confuse motion with action.”
13. “When in doubt, don’t.”
14. “It’s easier to prevent bad habits than to break them.”
15. “Three can keep a secret, if two of them are dead.”

I hope you enjoy this post. Please pass this onto others who may enjoy it!    Also, click “Like” below if you like it!