502: Powerful Question

It's not included in the PEMDAS Order of Operations ...

Should $a^{b^c} = ({a^b})^c$ or should it be $a^{b^c} = a^{(b^c)}$ ??

Does $3^{2^0}$ equal 1 or 3?
 
Let's just consider easy numbers {1, 2, 3, 4} so we can explore. What's the probability that the two methods arrive at the same answer?

For the record,  $a^{b^c} = a^{(b^c)}$ is the accepted order of operations here.

432: Stuffing Bags

Sam Shah wrote:

Matt Enlow (math teacher in MA) posted a fascinating problem online today, one he thinks of when storing all those plastic bags from the grocery store. You shove them so they all lie in a single bag, and throw that bag under the sink. Here’s the question: how many different ways can you store these bags?
For 1 bag, there is only 1 way.
For 2 bags, there is still only 1 way.
For 3 bags, there are 2 ways.

Here is a picture for clarification:

Can you figure out how many ways for 6 bags? 13 bags?

source.

408: Balls

source.

347: Combinatorics

Consider eight objects. We will choose them one at a time, two at a time, three at a time, and so on.

Which of these will result in identical numbers of ways?
Why?

346: Casting the Play

The cast of a school play that requires 4 girls and 3 boys is to be selected from 7 eligible girls and 9 eligible boys.

• Will it be a different calculation if the boys are willing to play girls' parts, as in Shakespeare's time? If so, how will it be different?

224: Combination (?) Locks 2

Which lock is more secure?

223: Combination (?) Locks 1

Which one is most secure?

Three numbers ...

This one with three extra wheels you can swap in ...

Or this one, with three letters?

171: Steering Wheel.

These are the control knobs on a Formula 1 Race car. How many can you figure out the purpose for?

131: Traveling Salesman

from MathGIFs:

This is a visualization of the Traveling Salesman problem.

"One of the most famous problems of math and computer science is the Traveling Salesman Problem. Given a list of n cities, what is the shortest tour that visits each city exactly once and returns back to the starting city? One of the reasons why this problem is so famous is that it is easy to understand the problem and incredibly hard to solve the problem. The animation above illustrates the steps a computer algorithm used to arrive at what we think is the shortest distance - 12,930 miles - needed to visit all 48 continental US capitals. (We used Google maps to use actual roads and highways when computing the distances between cities.) The best way to view the animation is to watch only one part of the US at a time, like the northeast. Then you can see how the web of paths works itself out into an efficient route."

Let's simplify things a bit ...

Find the shortest route through the capitals of the six New England states.

121: Train Revisited

What about the Sequel?

Check out AWESOME "how many ways can something be arranged?" from @afs8math http://t.co/P5dBwtDHau & http://t.co/qPR7Vqw6SU cc @mr_stadel
— Fawn Nguyen (@fawnpnguyen) April 15, 2014

120: Permutations - Arranging the Train

Stop the video before he starts giving out answers.

Well?

Check out AWESOME "how many ways can something be arranged?" from @afs8math http://t.co/P5dBwtDHau & http://t.co/qPR7Vqw6SU cc @mr_stadel
— Fawn Nguyen (@fawnpnguyen) April 15, 2014

100: Combinations. One more Twist.

We've seen this a couple of times now.

What if I had a white marker, one that didn't show, but still held a place?
How will that affect things?

Image from Mr. Stadel.

99: More combinations

Blue, green, yellow, and red ... We seem to be all set.

What if I get three black markers out of the drawer?
How will that affect things?

Image from Mr. Stadel.

98: Combinations!

Blue, green, yellow, and red ... We seem to be all set.

Now, what about the black, purple, and orange ones?

And BROWN!

Image from Mr. Stadel.

49: Pizza, pizza

Pizza ... makes you hungry doesn't it?

What if you had ten toppings available. How many different two-topping pizza variations can you make ...

What definition did you have to "adjust" in order to get David's answer of 55?

Do you agree that this is a fair interpretation of the problem?

source:

10 pizza toppings Choose any 2 - different or same; How many choices?
10×9÷2=45?
10×10÷2=50?
10×11÷2=55?
10C1+10C2?
Ans: 55 Why?
— David Marain (@dmarain)