Marsbarn Designs...

The locker problem, or more math thoughts

Yes, another 6th grade math problem:

The locker problem goes like this:

A school has 1,000 lockers and 1,000 students.
The first student runs through the school and
opens all of the locker doors.

The second student runs through the school and
opens every second locker.

The third student runs through the school and 
"changes the state" of every third locker. (This
means she opens closed lockers, and closes open
ones.)

The fourth student runs through the school and
changes the state of every fourth locker.

And so on, until the 1,000th student goes through
the school and changes the state of the 1,000th locker.

How many locker doors are open?  Why?

What do you think?  Scared? Intrigued? Curious?

The book recommended taking a small sample
and charting what happens as the students move
through the school. Using a 25 block grid, I went
a step further and wrote down the number of the 
student on each square as they touched a given locker. 

If you do the experiment, you find that the 
first locker is open (only the first student ever
touches it), the second and third lockers are closed,
with two students having touched each.  The fourth
locker is open.  The fifth through 8th lockers are
closed.  The 9th locker is open.  This continues.
Analyzing the grid, I find that only lockers 16 and 25
are open.  All of the others are closed.

So.  Lockers 1, 4, 9, 16, and 25 are open.  They are
all perfect squares.  What makes a square number
special in this case?  It's the number of factors.
A non-square number has an even number of factors.
(For example, 12 has the factors of 1, 2, 3, 4, 6, and
12.)  A square number has an odd number of factors.
(25 has factors of 1, 5, 25).  

Not only can you determine how many lockers are open,
you can figure out how many students touched each
locker.  If you choose a locker, say number 100, you
can determine how many people touched it by counting 
its factors.  100 has factors of 1, 2, 4, 5, 10, 20, 25,
50, and 100, so 9 students touched that locker, and
the locker is open at the end of the experiment.  

I would love to say that I figured this problem out
in no time flat, but I didn't. I got stuck.
My first idea was to use prime factors, and that seemed
to work, a bit.  Until I realized it didn't work at all.
This is where having a complete model, one with 
not just open and closed door symbols, but the student
number that touched the door written on the chart,
really helped.  Still, even with my misstep, I really
enjoyed the time I spent charting and figuring out
this problem. So. Much. Fun.  

Another custom dice bag

This is my latest custom dice bag.

I had fun with this one.  It's reversible, so I was careful
with the add-ons.  The face is all fusible vinyl.  The 
tongue (which I love) is cotton with a bit of batting inside.

The reverse was more of a challenge.  With its simple 
features, the face was difficult to give dimension to.

There is a layer of light grey vinyl under the blue vinyl 
of the eyes to give it some pop. I then pieced the grey
of the robot face so that it angled into side panels to 
give it more shape.

I was really pleased with how this turned out, but (in a
sad aside)  this poor little bag is currently wandering
lost in the postal system.  There was a mix up with the
address, and I'm now waiting for him to come back
home so I can send him off again.

a new math obsession

Okay. I wrote a post all about a math problem.
Really.  The whole thing.  
You've been warned.  If you don't want to read a lengthy post about 6th grade math, you should go look in the archives. 

I have a new obsession.

Perhaps I missed this trick in middle school,
or (as I suspect) I was never shown it,
but I've found this pair of techniques to be 
fascinating and (dare I say it) elegant.

In my continuing exploration of middle
school math both to be a better tutor and
because I find it addictive (everyone has hobbies),
I've been playing with prime factorization.
We called them Factor Trees in my day.  You know,
these things:

My own ancient memories of them were that they 
were pointless seeming exercises in breaking down
numbers into their bits and pieces.  But in this here
modern world of math, they do things with them.

This first trick, I vaguely remember.
Start by rewriting the factors like this:

If you take all the prime factors both numbers have
in common  (two 2's and a 3), and multiply them out,
you have 12, which is the largest number that can
divide into both numbers evenly, the Greatest 
Common Factor.

Okay, the second trick is for finding the Least Common
Multiple. It's the number both numbers (180 and 48)
can divide into. I remember writing long lists of 
multiples until I would stumble upon a number they
had in common, or just multiplying both numbers
together and dealing with HUGE numbers. 

But this is a slick, pretty, and (dare I say it) easy
solution. Look at the prime factors again,

This time you take all of the factors you 
need to make both numbers.  Think of this 
as a building a kit to build two different
lego shapes.  You can share blocks, but you 
have to have all the right shapes.

So, we need four 2's, two 3's, and a 5 to 
make both numbers.  If you multiply
these numbers, you get 720, 
which is the smallest number both 180 
and 48 divide into evenly. 

Why you ask?  (Okay, why I ask.)
The four 2's, two 3's, and a 5,  
can be combined like this:
(2 x 2 x 2 x 2 x 3) x 3 x 5 = 48 x 15

or like this:

(2 x 2 x 3 x 3 x 5) x 2 x 2= 180 x 4.

 I don't know why I find this so cool, but I do.  

How the cookie got there

Little e:  Mom where's my book?

me: Why would I know?

Little e: (blank stare)

Sweetheart G: "Because you're the Momma."

me: You're like 'Mom, where's that thing that looks like a dandelion, but is bent in half and is kind of blue but really it's red, and I haven't seen it since I was five?'  and you expect me to know!

Little e: Mom.

me: Yes.

Little e: Do we have something like that?

me: Um, no.

Little e: Oh, okay, 'cause if we did, I'd ask for it.

And that's when Sweetheart G tried to laugh cookie out through her nose.

Thinking about fall and beyond

We are having another run around with summer
but I know fall is just around the corner.  

With the school routine taking form, thoughts go to
what is next. In my head (where chores and limits
don't intrude), I'm thinking of two separate sets
of art/craft classes this November and December.
I want to have adult classes and kid classes where
gifts are made (either for yourself or for others).  
My favorite part of the pre-holiday season is the making
of gifts, and I want to share that. 

I'm thinking of silk scarf painting, felt ornaments,
collage, and vintage ornament making as possible
subjects.

Anything you want to make this holiday season? 

like the honey bees, we buzz

It's the end of the second week of the school year,
and we're just starting to find our rhythm.

Sweet G's middle school schedule runs later than it did
in elementary school.  She gets to sleep in (yay!), and
have a bit of free time in the morning (Yay, yay!), but
it also means she gets home much later.  With Big E's
work schedule, they completely miss each other 4 days
a week.  It's a big change, and a hard one.   

Yesterday, Big E had lunch with Sweet G in front of the
middle school.  It was a treat for both of them. It's not
an everyday, or even an every week, sort of solution,
but it's something special for them, which counts for
quite a lot. 

Our days are feeling full these days between two 
school schedules, Taekwon Do, homework, and math
tutoring.  But it's a good sort of busy, the sort of
schedule that a family could get used to.   

Silk scarf painting, or something totally new

For our Ladies Friday Art group, we tried something
new last week.  This was a seat of your pants, no 
dry run, 'just open the package and go' sort of project.

I'd gathered the supplies over a 2 year period (yes, 
another one of those "this looks cool, buy the supplies
now, but I have no time table in sight" type projects).

(Someday, in the distant future, I think Big E will be
nominated for sainthood based purely on his lack of
eye rolling when I come home with bags of art supplies
that have no immediate purpose.  He doesn't ask,
"And when are you going to do THAT?" He just smiles
and tries to look focused when I excitedly blather about
vague plans for future projects. Of course, the fact that
I don't querry him about his projects might provide a
sort of quid pro quo.) 

We used a water soluble resist applied with a needle
nose applicator.  Questions like "does the resist need 
to dry?" were answered through try it and see approach.
(The answer is no, but letting is set up a bit is good.)

The dye was an instant setting concentrated dye that 
is diluted with water.  No chemicals to add or mix,
nothing to dissolve or measure precisely. 

The dye is delived via dropper bottle or pipette.  It's 
a bit odd to work with, because the dye doesn't flow,
it soaks through the silk.  The silk almost pulls it 
from the delivery device. 

The resist worked beautifully.  There was little or no
problem with the dyes running into each other.  I had 
worried about one of those nightmare watercolor
incidents where everything runs together, but no.
These dyes are (dare I say it) almost obedient.  

The dyes could be combined to create new colors, and 
the concentrate really went a long way.  When we 
were done painting our silk, we had to let the scarves
sit for an hour to dry.  After an hour, I washed out the
resist and excess dye.  It was the easiest clean up I've
ever had.  Just a bit of running water, and *poof*
finished scarf.

No boiling, no steaming, no fuss, no muss.
So. Much. Fun.    

The first two photos were scarves in process, the
last five photos were of our finished scarves.  We 
worked on top of towels which had been placed on
a plastic covered work table (okay, dining table).
When we lifted (peeled) our dried scarves from the
towels, a beautiful shadow of the scarf was left behind.
The echo image turned out to be permanent, so now 
we have beautified utility towels. 

I have all sorts of new ideas for this process.  I think
it will be the perfect indoor dyeing project for the
winter.  (Because, you know, I was lacking one.)

calculating new forms of insomnia

You know those nights when you cannot fall asleep?  
Your brain won't shut off, it just hums along with
lists and To Do's and budgets and worries.

Well, last week, I had a night when my brain would
not quiet, but instead of Big E's work or the To Do
list keeping me up, I kept thinking about....
math.  Yes, math.  

I thought, "If you bought a dozen items that each
cost 99 cents, how many different ways could you
calculate the total cost?"

For about ninety minutes, everytime I would come
close to drifting off, I'd think of an alternate 
method.

First I thought of these two: (You can click on the calculations to make them bigger.)

Then, I moved on to these two:

Seriously. I'd almost drift off, and then this...

Now the idea how to carry a "ten" became particularly
fascinating as the evening moved into morning.*(If you
have any interest in the rambling math thoughts of the
sleep deprived, see the footnote.) 

But the brain wasn't done... no, five methods weren't 
enough.  I thought, if I were to do this in real life,
I'd do this:

So, sometime after 12:30 AM I drifted off past the
math delusions into a very short night of sleep.

Total crazy pants.  Absolute, complete craziness.

 

 

 *Do you carry "ten" into the "tens" column, or do you carry
"one" into the "hundreds" column?  Of course you have to
carry twice, when you count up the "90"'s, you again get
that choice to carry "ten" into the hundreds or "one" into
the  "thousands".  Which leads to the thought of combining
the two carries into "eleven" carried into the "hundred"'s. 
Do these all count as seperate methods?  Who freakin'
knows!   

One more fine day

It seems hard to believe that the summer
is over. *Poof* We are back to our regularly
scheduled program.

I mean, we were just there, right?  In vacation land,
and now here we a week into another year of school,
and volunteering, and alarm clocks, and school lunches.

And the choices.  Here we are making the choices.
Do I do this, or that, or the other thing?
What are the commitments I make for this year?
How much at school, how much at home, what kind
of work?

Of course, there is also the learning and the growing,
and without the work, we don't appreciate the time 
off.  But still,  another day at the beach sounds
awfully Fine, right about now.  And yes, that's "Fine" with a capital "F".

Batiking: Color Play part 4

On the day of the workshop, I did no actual
batiking myself. That's not what the day was
about.  But, the following week, I spent a day
dyeing the last of the workshop fabric,
and working on some fabrics of my own. I don't
know if I'll do any more dyeing before fall sets
in (ie it gets too wet to work outside),
so I wanted to finish up any on-going fabrics.

This fabric is one I started in June.  I started with white,
dyed it magenta, added hand painted stripes of wax
and then overdyed it blue.

This is the other end of that fabric.  Yes, I was having
fun, a lot of fun.  

This fabric is another that I'd started in June.  It's a 
different base cotton which took the wax a bit
differently.  I started with a pink base, painted on
stripes, and then boiled off the wax, but the wax didn't
come out all of the way.  And, I wasn't happy with
the pink/purple color.  So I put another layer of wax,
painting my stripes 90 degrees off from the first layer
of color, and dyed the fabric red.  When I boiled it for
the second time, I made sure all of the wax came out.
I really like the variety of tones in this piece, but this
cotton has a much heavier hand than the other cotton
I used for class.  

A second photo of the same fabric.

Ah, this last fabric.  I (heart) this fabric.  By the end of
my day waxing and dyeing, I wanted to break out of the 
box a bit.  Enough with the dyeing complementary 
colors. I wanted something interesting.  So I took a 
turquoise base color (a lovely color by the way), 
waxed it with a wooden block, and dyed it orange.

Here's a bit of color theory, what do you get when you
mix orange and turquoise? With my dyes you get this
funky cool combination of turquoise, green, brown, and orange.

So. Much. Fun.  Now, I need to start sewing with my
new collection of fabrics.  Quilt anyone?