Ok, so who's good at math?

Fenton got this cute little penguin puzzle for Christmas, which the whole family has taken a liking to. Some days Fenton likes to dump all the pieces and stack them back up again. Other days he likes to dump all the pieces and use the base (that has a wooden rod to hold the pieces in place) and bang away like it's a hammer. While it does make a good hammer, I wish he wouldn't use it to pound imaginary nails into my knee. Kids and their imaginations these days!

But the rest of us feel the irresistible urge to see how many different ways we can arrange Mr. Penguin. Indeed, it seems that every time I walk by him, he's a changed man.

So this got me wondering: should I vacuum today or tomorrow? Oh, wait, that's silly of me. Why was I wondering that? Of course I need to vacuum both days. Pshaw! As if!

But I was also wondering: just how many ways can we arrange Penguin Jr? As I spent some time thinking about it, and consulting with Damon, I quickly realized that this is not only a complicated question, but it just might require that thing called math! [gasp!] I know, I know! I wasn't ready to face that fact either. I know I need a break already, how about you? Yes? Ok, break time. So everyone watch this video and then come back.

Feel better? Yeah, Star Wars dogs rock!

Ok, let's see if we can tackle this, shall we? Our mission, should you accept it, is to see how many ways we can arrange Senior Penguin.

So let's start out with the facts. Just the facts, ma'am. Ok, there are 7 pieces to the Perky Penguin puzzle - not including the blue hammer/base. Let's leave that out since it doesn't really factor for possible combinations, since it can only be on the bottom (or flipped upside-down to be on the top).

Are you still with me? It's going to get even more complicated in a minute, so if you need a short nap, I understand. In fact, let's just all go ahead and take a little siesta.

[Time passes... Snoring... Wiping off drool... Massaging keyboard prints off forehead...]

Oh, are we all back? Ok, good. Now where were we? Oh yes. So how many ways can we arrange The Penguinator's 7 pieces?

I think we need to think about what we are really asking here. More specifically, are we asking if it would be better to order a pizza or just make macaroni and cheese? Both have their advantages, obviously. Pizza = easy. Mac N Cheese = cheap. Oh, why can't I get Mac N Cheese delivery? Woe is me...

Oh, right. Sorry. My husband think my son's ADHD tendencies may come from my side. I don't really know what he's talking about. So have you solved the problem yet? You know, the Persnickety Penguin problem?

No?

Ok, fine. I guess I have to sit here and walk you through it. Did you understand the part about having 7 pieces? Yeah, ok good. So thinking about all of this, and searching through the Land of the Internet, I realized that if you type in "Gingrich" the first hit you get is "Mitt Romney for President." No joke...

Well played, Romney, well played.

I also realized that what we are dealing with here are permutations (not combinations). Permutations focus on the arrangement of objects with regard to the order in which they are arranged, whereas combinations focus on the selection of objects without regard to the order in which they are selected. For example, consider the letters A and B. Using those letters, we can create two 2-letter permutations - AB and BA. Because order is important to a permutation, AB and BA are considered different permutations. However, AB and BA represent only one combination, because order is not important to a combination. (Thank you, StatTrek!)

Got that? I just threw a whole lot at you. Sorry about that. Do you need another break? How about before we go on, we get a little refresher on math? I think we'll feel a bit more confident in our selves if we do so...

Oh man, I don't know what to say. I mean, math is really pretty easy after all, isn't it? I feel so much better.

Let's continue.

If we have 7 pieces, how many different ways can we arrange our Wacky Waddler? It turns out we need to work with something called a factorial (don't ask me how I know, I just know). Say it with me... ffffaaaaaccctttooorrriiiaaaalll. Good! A factorial really is just a fancy word for "a non-negative integer n, denoted by n!, that is the product of all positive integers less than or equal to n." Who knew? And I thought it was complicated math term.

For example, the factorial we'd use for our 7 piece puzzle would be:

7! or (7)(6)(5)(4)(3)(2)(1) = 5,040

Great! Wow, 5,040 is so many combinations.... err... sorry... permutations. I'm so glad I didn't try to take a picture of each one.

Oh, but I still am wondering something... No, it's not about vacuuming or pizza. Come on people, stay focused! What I'm wondering is what if we turn some/all of them upside down? The factorial we just did only is if everything is just one way. But we can have all facing up, all facing down, or 1 up/6 down, or 2 up/5 down, etc.

Wow! Ok! So what does that mean? Well it means things just got a whole more complicated. Yeah, I know... Need another break? Ok...

Back? Good. Let's think this through... So there are 5,040 permutations with all facing up. And there are just as many permutations with all facing down. And I'm guessing that each up/down set would also have 5,040 permutations. If I'm thinking this through right, it would be something like the following, since each piece can ONLY be up or down:

5,040 permutations of 0 up/7 down
+ 5,040 permutations of 1 up/6 down
+ 5,040 permutations of 2 up/5 down
+ 5,040 permutations of 3 up/4 down
+ 5,040 permutations of 4 up/3 down
+ 5,040 permutations of 5 up/2 down
+ 5,040 permutations of 6 up/1 down
+ 5,040 permutations of 7 up/0 down
--------------------------------------
= 40,320

Ahh! So our answer is 40,320. Excellent work everyone.

What? Say that again? That's not right. Almost right? What!!!

Ok, let me rephrase what I'm hearing you say: That's not taking into account each piece. Let's take the head, for example. The 1 up/6 down scenario works if you just take one piece, say the head, and use it as the only "up." But you could have just as many permutations if the "1 up" is the feet. Or the black bottom. Or the weird rubber flippers. Etc. And then for the 2 up/5 down, you'd have to do the permutations for each combination of twosies/fivesies.

Oh man, oh man, oh man! What have we gotten ourselves into? Is that a vortex opening up? I think I need to lie down.

[Lying down...]

Oh much better.

What? You still haven't solved it? Do I have to get back up? Fine.

[Getting up...]

It's obvious at this point that we are also working with event multiples. Yeah, I didn't know they existed until about 3 hours ago. Apparently it would work something like this:

We have 7 events (items), and each of those events can be either of two ways (up or down).

When you calculate the event multiples for this, you get 128. I can't really say what the formula is to calculate that, but trust me...truuussssssst me...I know what I am talking about. (Also, StatTrek has a neat calculator - go ahead, try it. See?)

Ok...so what does 128 have to do with the 40,320 we came up earlier? Hmmm....sounds like break time!

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All right. I know I feel better. Ok let's get this finished up. Re-runs of Three's Company are about to start and I can't have math on my mind for that. Are you with me? Who's with me!? Oh, you are. Good.

Ok, so I think what all this crazy stuff means is that there are 128 ways we can stack our lovely 40,320 Pretentious Penguin permutations. Does that sound about right? If I'm right, then there are 5,160,960 different ways we can arrange Papa Penguin.

Let's say that again (just for fun): 5,160,960. Or as Damon puts it, a ridiculously* huge number. Yeah, huge.

Now if I haven't thought this through right, and you think my calculations are off, please comment. I mean, really, what the hell do I know. Math is hard.

. 
* You know what he really said.

A kid's view of paradise

I had a rough childhood.  I was forced to live on a desolate island in the middle of nowhere for years.  I had to suffer through extreme hot and cold temperatures ranging anywhere from 72 and 74 degrees.  I had no choice but to go sailing or to the beach every weekend.  I remember the burden of climbing palm trees and eating fresh coconut, or having to hike up trails just to see a waterfall.  Yeah, it was a tough experience, but I know that it has only made me stronger.

But now that I am grown up, and now considered a 'mainlander' by many, I occasionally think back to my years as a child in Hawaii.  My cousin just got married and went there for her honeymoon and she sent some pics of the house I used to live in, so I went ahead and dug through my pics from when I was a kid and thought I'd post some of them.

Of course, this was back in the days of film, so you know I have a total of 67 pictures from my five years in Hawaii.  I know my parents have more, but I'm just working with my stash that I scanned a while back.

So yes, let's take a tour, why don't we? 


This is me at the beach.  Annoying sand and sun everywhere, which I had no choice but to play in all day. 

This is me doing a cartwheel in a friend's backyard.  I was really good at cartwheels.  Apparently, this is one of those 'use it or lose it' skills. 

This is the side of a cabin we stayed in when we visited one of the other islands.  One time, when we went to some other islands, we rode on a military ship because it was free or cheap or my dad thought it would be fun or something.  It was hot on the ship as there was no shade.  It may sound neat, but when you're floating at sea with the tropical sun pounding down on you, it can be a little annoying. We all kept swaying back and forth for like a day after we got off the ship.  It actually was a cool experience, looking back on it.  I don't think that this trip was the 'ship' trip though.

This is the door of the cabin...because, you know, the only thing better than a picture of the side of a cabin is a picture of the door. I think this cabin had a fireplace, because I guess in some places - even in Hawaii - it gets cold sometimes.

This is a picture of some cows crossing the road.  I had never seen anything like it before.

This is another picture, but mostly I got my dad's camera in the shot. He gave me the camera years later and I still have it.

I guess this is some field we went by.  I can't make out what is in the distance.  Some cliffs maybe?

We visited the world's largest macadamia nut orchard. Macadamia nuts are pretty yummy.

This is a pic of one of our times out sailing, I think.  I suspect that we had seen dolphins or something, as it looks like one has just gone back under (in the bottom left of the pic). It's hard to tell, so I'm probably just making things up.

There were lots of places we would drive by that had black rocks - probably lava rocks - on the side of the road.  This place also had white rocks for some reason, and people would stop and spell out cool things.  I spelled my name.  I thought I did a nice job.  I'm sure it's still there, too.

This, I believe, is a volcano crater.  I lived there back before the volcanoes became active again, and we actually walked across one of the craters that is now active today.  I'm not sure if this is Kilauea, but the pic further down is.

This is me, standing in the crack of a road that we had to hike on to get to wherever it was we were going.  Judging by the order of these pictures, I'm guessing it was on the way to the crater.  But it's also possible that this was taken months later, so who knows.  And you think you've got bad pot-holes where you live.

More hiking - probably to another (or the same) crater. I guess the bulge in the land is due to the ground moving from a previous lava flow or eruption or something.  I don't know, really.

Ok, so this is Kilauea*. If I was anything like my kids, I'm sure I complained constantly during this hike, not appreciating for a second that I was walking over a volcano that would begin erupting again one day. 

*Although my pic is a bit blurry, it does look a lot like this pic on Wikipedia of Kilauea at Hilo:

This is a picture of a car we rented on one of our trips.  I have no idea why we pulled over to the side of the road.  Maybe to look at something?  I dunno.

Maybe it was to look at the pretty flowers.  It was quite pretty there.  It's also possible it was a pee break.

I remember seeing topiaries like this a lot there. This one had all kinds of zoo animals.

Ahh, yes...the waterfall.  Another tortuous hike, I'm sure.  Climbing up flowery, wooded paths, with the birds calling and roar of the water in the background.  I know I made it as miserable for my parents as possible.  Sorry about that.

My mom used to make homemade popsicles that were awesome.  Full of sugar and fun to eat.  I also liked to pretend I had an enormous tongue.  I know, I was strange.

Speaking of strange, I also liked to take tennis balls to...err...enhance my figure. I really have no idea what I was thinking. 

This is our front door/carport.  I took a picture of one of my dolls, sitting at the bottom of the step. I liked that doll because it's face and hands were made of soft plastic, and you could pull down the back of her outfit to wipe her butt (also soft plastic).  I though that was cool.  Our Oldsmobile Cutlass was pretty cool too.

I'm not sure really what this is, other than cows in a field.  Let's mooooove on. (Ahem...sorry...couldn't resist)

Oh...more cows in the field...alrighty then...

This is someone's elbow, and also a neat bridge thing.  It's possible they were doing maintenance on it, since I see a crane or something. I have no idea why I took the picture, really.

This is the view from somewhere of something.  I think I liked seeing the farm plots and the stream in this cute valley in what seemed like the middle of nowhere. I wonder what they were farming.

Here is a second picture of it.  Considering film was a precious commodity to be used very scarcely back then, I must have really liked this to take two pictures of it.

Some picture of a mountain, taken from the car.  I can tell this wasn't our car, because our Cutlass was two doors (with a dark green interior), and here you can see I have my own lock on the door that I could control.  Mmmmwwwhhhaahaaa.

Yes, another waterfall.  Seriously, they were everywhere, often in random places.

And another one.  A double one, even!

Two waterfalls, two pictures!

Ahhh...telephone wires mixed with the defrost lines on the back windshield. I actually think it's kinda cool, even though I'm sure it's not what I intended.

Finally, a picture at the marina.  Must have been a sailing day.  One time I went sailing with some people and had to pee sooo bad that my Mom and I raced to the bathroom after we got back and I made it all the way there without going, but couldn't get my pants down in time so I sat on the potty and peed through my pants.  True story.

Someone forgot their anchor.  I'll watch it for them until they get back.

I just noticed what awesome shoes I have on.  If I had them today I'd be what you call a 'hipster.'  Of course, if I had those pony tails today I'd be what you call a 'dork.'  A dorkster?

Boats on the dock.  Probably the boats we would be using that day. Look how yummy-delicious that water looks!  I'd jump in right now if I could.

You can't tell, but I think that my brother is sailing that boat.  Yeah, he sailed too.  So did my mom.  My dad could also sail.  We all knew how to sail.  It was like riding bikes in our family, except that I didn't know how to ride a bike, and it was probably more expensive.  And ropes and sails were involved.  I'm sure there are some other differences, too.

Yeah, that's my brother.  I'm not sure who he's sailing with, but it reminds me of a joke my brother told me that his sailing buddy told him about the Purple Gorilla. He said it took 3 hours to tell the joke.  I can tell it in about 3 minutes.  Amateurs.

I was never good back then (or now) about waiting until people moved out of my shot, so I now forever have a picture of some lady looking at some cool mountains.

Oh, that's better.

One time I got to take a tour of a submarine, and we all got a picture to take home.  This is the para-scope, and I had just looked through it.  I remember thinking "Man, you really can't really see shit through these things."

My brother would build these crazy things with the broken tree limbs in the backyard.  I tagged along with whatever he did.  Here I am wearing my Midway shirt, which was one of two shirts I wore as often as my mom could wash them.  This was my blue outfit - the other being my red outfit.  I wore that shirt forever, and there is a good chance I still have it somewhere.

These next few pics are of some harbor (quite possibly Pearl Harbor) that we are looking down on as some ships come in.  I remember it being very windy.  My dad was in uniform there, so I think there was some event going on.

Oh, and here are the pics my cousin sent of what our house looks like now. What strikes me the most is how all the trees are gone.  We had coconut trees along the right with a grass side yard with a lime tree.  All that's paved over now.  Also, we had a whole bunch of trees and plants in the front and left side of the yard - now all gone.  Those pictures earlier of me with the popsicle and the boobies were taken in our front yard.  And from what I can see, the back yard has been cleared since I can see that tank in the background which was blocked by all the trees before.  Much different than what me and my brother would play in.  But I guess it looks the same otherwise.  That bedroom in the front (on the left) was my room.  Our house had this cool wall mural inside that someone painted before we bought it.  It had a rainbow and a turtle and some other stuff.  I wonder if my parents have a pic of it.  It was pretty neat.

Well, folks, I'm pau.  Huh?  What?  What is 'pau'?  Oh, yeah, sorry.  Speaking of which, have I ever told you my 'pau' (pronounced 'pow') story?  In the 9th grade, I was in the bathroom with some girlfriends and I asked them if they were pau.  They had no idea what I was saying.  "You know...All done? Finished? Pau?"  My questions were only returned with blank stares.  The fact that I only realized this was a Hawaiian word when I was in high school - and had been using it regularly up till then - probably accounts for some of my uncoolness that lingers to this day.  ("Can you believe she said 'pau'?") Yeah, I'm sure that's it.

Anyhow, I did leave out some pics that had my family in it, sparing them the embarrassment of being seen on the web with 70's hair styles (you're welcome).  Maybe some day I'll post some pics from California, although I have nearly 300 pics from that era, and even more from Maryland, so I'll have to pick through them for the gems.

A hui hou.