Z is for . . .

The final Mad-Cool-Math Nugget is . . . the Zollner illusion.

Image courtesy: Wikipedia

Besides making you go cross-eyed, the purpose of this optical illusion is to make your brain doubt that the diagonal lines in this image are parallel. The shorter lines set on angles mess with your perception. This illusion is named for the person who designed it: German astrophysicist Johann Karl Friedrich Zöllner. (Wikipedia)

Want to see more? Go to this Wikipedia page. I grabbed a few of my favorites to share here.

In this Simultaneous Contrast illusion, the background changes from dark grey on the left to a lighter grey on the right. Believe it or not, the bar in the middle is one shade of grey. Yes, really.

Image courtesy: Wikipedia

In this next one, focus on the black dot while you move your face closer to and then away from the screen. Don't go too fast. We don't want any broken noses. The outer rings should appear to rotate. Trippy.

Image courtesy: Wikipedia

And finally, the dancer. Is she moving clockwise or counterclockwise? The first time I saw this she seemed to switch direction after a second and danced clockwise from then on. I keep trying to see her dance counterclockwise, but my mind isn't cooperating. Typical. (I was also one of those people who saw "The Dress" as cream and gold.)





By Nobuyuki Kayahara (Procreo Flash Design Laboratory) [CC BY-SA 3.0], via Wikimedia Commons

And that's all folks! Sing it with me: Now I've blogged my ABCs, next time won't you blog with me!

 It's been a blast. Thank you to all who've stopped by, commented, followed, and maybe even made off with a free download of one of my stories. Special thanks to those bloggers who kept coming back for more despite the math theme, including

Chrys Fey at Write With Fey

Nick Wilford at Scattergun Scribblings

Cold As Heaven

Rena Rocford at Doctor Faerie Godmother

Lynda Grace An Hour Away

Sarah Foster at The Faux Fountain Pen

Keith's Ramblings

Mark Coopman at Viginette's from VA (and D.C. too!)

Alex J. Cavanaugh

If I forgot anyone, my sincere apologies. See Read you next week at the IWSG!

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Finally, if you missed any of the free downloads from my new collection Ursa Major And Other Stories, the price is reduced today through the month of May.

Y is for . . .

The Y Game

Image courtesy: Wikipedia

In this game, two players take turns laying chips on a triangular board. The first player to make a path that connects the three different sides wins. The winning path will form a twisted version of the letter Y, hence the name. I probably should say something about strategy since my theme this month is Mad-Cool-Math Nuggets, but nah. Not in the mood. Yesterday's blog was sufficiently mathy for several days worth of blogs.

This game caught my eye because my kids have a slightly different version. In Bingo Link you only have to forge a path between opposite sides of the hexagon. The sides are color coded red, blue, and yellow so you don't forget where you are going. The twist is that each player calls out the picture in the spot they choose to place their marker. The other players then have to find that image on their board and they get to place a marker in that spot. Maybe it could be called the I-game, for I-spy.

Image courtesy: Amazon

Bingo Link is quick to learn, but boring once you're past age 6 or 7.

What are your past or current favorite board games? This past Christmas we got Forbidden Island which is a great cooperative board game. We also love One Night Ultimate Werewolf. That one looks more like a card game than a traditional board game, but I highly recommend it.

X is for . . .

This month I'm posting Mad-Cool-Math Nuggets to foster an appreciation of all things mathematical. Then I'm going to teach pigs to fly.

Image courtesy: Wikicommons

X is for . . . the following problem:

Let x = 0.99999 and don't ever stop typing nines, because they will be repeating for all eternity. Wait, we can't do that. How am I going to get the proper notation on Blogger? Usually, you draw a horizontal bar on top of the first 9. But that's going to be tricky. Some places uses parenthesis, such as 0.(9). Ew.

Figulty fum. Let's just agree that 0.9999repeat will be my notation for this repeating decimal today.

So now, I will amaze you by proving that 0.9999repeat is actually equal to 1! Better than pulling a rabbit out of a hat, huh?

Here's how it's done:

Let x = 0.9999repeat

Multiply both sides by 10. In algebra, equality will hold as long as you perform the same (legal) operation to both sides of the equation. Something not legal would be division by zero. Note that multiplying 0.9999repeat by 10 looks like giving this number a little push to left. The decimal is now after the first nine instead of before it.

10x = 9.9999repeat

Now we will write 9.9999repeat as the sum of the following two numbers: (Don't freak out, this is equivalent to breaking 1.5 into the sum 1 + 0.5. No biggie.)

10x = 9 + 0.9999repeat

Next, we will perform a substitution. Since x = 0.9999repeat (go back to step one if you forgot), it is perfectly valid to rewrite the above as:

10x = 9 + x

Are you still with me? Good! Now we will subtract x from both sides of the equation. Does that seem strange? Don't worry, it will make sense in a moment.

10x - x = 9 + x - x

Do you know what 10x - x is? (Just say: 10 rabbits minus 1 rabbit leaves me . . . 9 rabbits!) So, 10x - x = 9x. Along the same vein,  x - x is zero. Nothing. Which means x - x can disappear, like magic. So the above simplifies to:

9x = 9

Now we will divide both sides by 9:

(9x)/9 = 9/9

Now 9 divided by 9 is 1. (FYI, we don't write 1x, because it looks weird. We just write x.) So on the left, the 9s "cancel", leaving us with x. On the right, we have 1. So

x = 1

Wasn't that great! Don't you love algebra! Happy dance! We started with x = 0.9999repeat at the beginning and after 3 different operations to both sides, 1 substitution, and 1 rewrite of a number into the sum of its parts, wah-la! We end up with x = 1.

Or is your reaction more like this fellow's?


Image courtesy: John Benson

No worries. The remaining Mad-Cool-Math Nuggets will be extra light. I think we may all be experiencing blog fatigue.

W is for . . .

This month I'm posting Mad-Cool-Math Nuggets.


 Image courtesy: rsteup

W is for Wald, as in Abraham Wald and a memo he wrote during WWII about damage to planes that had flown over Germany and returned. The idea for this blog comes from the article Five Statistical Problems That Will Change The Way You See The World from the site Business Insider.

As a member of a statistical research group, Wald's job was to collect data on which parts of a plane were hit the most by German fire. This information was supposed to help determine where the planes might be reinforced. So far, so good?

Now Wald found more bullet or flak damage to the fuselage and fuel systems than to the engines, so guess which parts he recommended to reinforce?

The fuselage and fuel system, right? No! Here's the twist: He was collecting data from planes that had returned, not the ones that were shot down. So Wald reasoned that the fuselage and fuel system could handle serious damage from bullets and return, while the engines could not. In other words, the returned planes showed little damage to the engines because the engines that did sustain serious damage never made it back.

Smart fellow.

This kind of statistical research in WWII was the beginning of Operations Research, which uses techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, and mathematical optimization, to get optimal or near-optimal solutions to complex decision-making problems. (Wikipedia) That was a mouthful, wasn't it?

Here is some information about the plane in the image above from the photographer:

A beautiful C-47 at the 2009 Geneseo, NY air show. This aircraft originally served with the 12th Air Force in the Mediterranean Theater in 1943 and the 9th Air Force in England 1944-1945 as part of the 316th Troop Carrier Group. It was one of the lead aircraft of the first strike of the D-Day invasion on June 6th, 1944. More info at 1941hag.org.

Impressive! Coincidentally, Geneseo is practically next door to where I live, Rochester, NY.

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You can download my novella, Ghosts of a Beneficial Place, for free today. It's about a mom juggling the stress of family, job, and a child with autism.

V is for . . .

This month I'm posting Mad-Cool-Math Nuggets.

V is for Venn diagrams which are named for John Venn who dreamed these up around 1880. Today these pictures showing relationships between a finite number of sets are used in probability, logic, statistics, computer science, linguistics, and humor. (Wikipedia) Versatile little buggers, aren't they?

Here's your basic, 2 set model with a nonzero intersection (well, kind of):


Image courtesy: Mike Atherton

Each circle represents a set. Set A contains people who are alive. Set B contains people who are pushing up daises. The intersection (those alive and dead at the same time) contain zombies. But you knew that already.

Here is a Venn diagram with three sets. (Those of you who might take offense at irreverent humor, please skip past this image. If I could blot out the word "mindless" I would.)

Have you ever thought of the connections between popular supernatural beings in fiction and religious supernatural beings before? It's kind of interesting actually.


 Image courtesy: Frantisek Fuka





Okay, you're past the potentially offensive part. Keep in mind that sets do not have to intersect at all.



Image courtesy: Bernard Goldbach

It is also possible for one set to be entirely contained in another. Let A be the set of books you wrote. Let B be the set of bestselling books. In a perfect world, the circle represented by A would sit completely inside the circle represented by B. For those of us that are still aspiring, the intersection of A and B remains empty. Alas.

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Interested in a free story with a hint of the supernatural? You can download Ghosts of a Beneficial Place through Monday for free.

I got the idea for this story from biking by a gazebo sitting next to an abandoned restaurant and thinking about how a haunting in books and movies is usually scary. The haunted place is the site of a murder or perhaps an insane asylum. Strong negative emotions stick to the place like glue. Well, could you have a beneficial haunting in a place where the emotions experienced were ones of joy? Where would that place be? How about a gazebo that held hundreds of celebrations like weddings, baby showers, and birthdays over the years?

U is for . . .

This month I'm posting Mad-Cool-Math Nuggets to foster an appreciation for all things mathematical. (Because becoming a published author of fiction wasn't challenging enough. (Insert sarcasm mark here. Did that ever happen by the way?)) Am I using parenthesis way too much? I think yes.


 Image courtesy: Ludie Cochrane

Okay, U is for unknown or variable. Is it just me, or should eighth grade algebra be renamed: The Quest for X? I mean, that's all I remember. Solve for x. Find x. Quadratic equations. Word problems. Every single time there was a variable whose value was unknown, we called it x. Maybe we should have used u for unknown or v for variable, but we didn't.

Why x? Well, it's not used as much in English compared to other letters. Using "a" could potentially be confusing since "a" is also a word. So what's up?

To the internet!

Guess what? There is a reason we use "x"! According to this article by LiveScience , an ancient Arabic algebra text from 820 A.D. used the word "thing" for variable. For example, 4 things equals 20 for scholars waaay back when would become 4X = 20 for kids today with a solution of "things" being 5 or X = 5 (because 4 times 5 is 20).

The Arabic word for "thing" was šay'. (Don't even think of asking for a pronunciation.) In the translation to Old Spanish, šay' became xei, which was later abbreviated to "x". Aha. Mystery solved.

Seen this image? You can get t-shirts or posters bearing this joke.


Image courtesy: Brandon I.

As a math professor, I would have viewed this as pathetic. The student just didn't study. But as a writer, I gotta give the student credit (or partial credit at least). The problem did not say "solve" or "find the value of" ; it said "find". Word choice matters, peeps. Am I right?

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From today through Monday, you can download my novella, Ghosts of a Benevolent Place, for free. Here's a blurb:

Having an autistic child is plenty challenging. Now throw in a teenager, a perpetually traveling husband, and a full time job. Audrey Ericsson has it all and then some. One day, by a little gazebo on a forgotten stretch of Lake Ontario, Audrey meets Gloria, whose husband Winston suffers from Alzheimer's. The unlikely friendship that blossoms between Ian and Winston is nothing short of miraculous to Audrey. When Ian begins to spell out words with rocks on the beach, Audrey is first thrilled, then puzzled, and eventually frightened. Are the mysterious stone messages from Ian, Winston, or something else entirely? (Approx. 74 pages)

T is for . . .

This month I'm posting Mad-Cool-Math Nuggets.

T is for tangent. Writers and/or their characters go off on tangents all the time, meaning we pursue a somewhat related or irrelevant course while neglecting the main subject. (The Free Dictionary) Take my character, Audrey Ericsson. She goes off on tangents all the time, but it's too be expected since she suffers from ADHD. And her son has autism. And boy, this has totally morphed into an advert for my new short story.

See what I did there? I went off on a tangent.

So what's a mathematical tangent? A tangent line touches a curve at one point without cutting across the curve (at that point). Imagine I ask you to balance a long spike on your head. Hopefully, you would opt to lay it somewhat horizontally on top of your head and not stab it into your skull. Same idea.

A tangent line can and often does intersect a curve at some other point as seen below. (Math is fun.) That's the source, but also a true statement. For some people. People like me who like math and can't stop going off on tangents. Gotcha!

Image source: Wikicommons

In this image, the red line is tangent to the black curve at the red dot (not where the red line crosses the black curve).

Lines can be tangent to circles too. Circles can be tangent to other circles. Planes can be tangent to spheres. It all happens the same way: the two items in question touch at one point.

Keep in mind that you can draw a tangent line at any point along a continuous curve. The video below shows many tangent lines to various points along a curve. The green lines are tangents with a positive slope (going up), the red lines signify a tangent with a negative slope (going down), and the black lines happen when the slope of the tangent line is zero (also known as a horizontal line.)

Is it just me, or does this look like someone riding a roller coaster while carrying a vaulting pole? Wheee! And speaking of roller coasters . . .




Source: By en:User:Dino (English Wikipedia) [Public domain], via Wikimedia Commons

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If you want to know more about Audrey Ericcson, stop by from tomorrow (4/24) through Monday (4/27) for a free download of the novella Ghosts of a Benevolent Place.