1 + 4 x 2 = ? .. 10 or 9 ?

[Wayne]

Here are three bent arrows.
You read the story from left to right, just like a normal sentence or equation.

What is the next bent arrow?

Choose from A,B,C, or D.

Say why.

Hint, break the story down into smaller chunks.

C
1st to 2nd is rotated clockwise 90deg and then mirror along the arrow line
2nd to 3rd is the same
so 3rd to 4th has the arrow facing up and flip from the D image to the C image.
Easier to show than write.

[McDuck]

C
1st to 2nd is rotated clockwise 90deg and then mirror along the arrow line
2nd to 3rd is the same
so 3rd to 4th has the arrow facing up and flip from the D image to the C image.
Easier to show than write.

Yes, a perfect answer.
You very accurately observed that there are two transformations between each step.
A rotation and a reflection.

[McDuck]

What's the next number in this sequence
and why.


1, 2, 3, 5, 8, 13, ?

[Wayne]

What's the next number in this sequence
and why.


1, 2, 3, 5, 8, 13, ?

Why so easy?
21
1+2=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21
ie ? = addition of previous 2 numbers.

[McDuck]

Why so easy?
21
1+2=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21
ie ? = addition of previous 2 numbers.

Any idea what that series is called, and why it's so special?

[Wayne]

Any idea what that series is called, and why it's so special?

Fibonacci Sequence though the Indians had found it prior to Fibonacci.
Creates the Fibonacci Spiral like the sea shell and other natural phenomena.

[McDuck]

Fibonacci Sequence though the Indians had found it prior to Fibonacci.
Creates the Fibonacci Spiral like the sea shell and other natural phenomena.

Yes, very nice.
Story goes that Fibonacci counted breeding rabbits.
And this is the way the population grew.

It's very different to how a dividing cell population grows.
Somewhat slower.
Begging the question, "In what way is the growth constrained?".

There is another relationship between the numbers in this series.
But it doesn't reveal itself until the numbers get a little bigger.
Since Fibonacci was such a star, I thought a little nod to him would be in order.

[McDuck]

Usual trick.
Break it down into smaller steps.
If you ever did logic or logic gates at school, that will be the way.

This is pretty tricky, there are four steps in between each frame.

[McDuck]

For this one there are two sets of rules.
A rule for dots, and a rule for lines.

Take the first picture (upper left) and overlay it upon the second picture.
Use the results to create the third picture.

With the dots, it's an ADD function.
"If the dot appears in the first picture AND the second picture, put it in the third picture"

With the lines it's a SUBTRACTION function.
" if the line appears in the first picture and second picture, subtract it from the third picture".

So following those rules.
The dot is gone, because it's not the same in either of the first two pictures.
And two of the three lines overlay each other, leaving only one line in the last picture.

So, by those rules, the answer is "F".

[McDuck]

Solve and show working.
Slightly harder.

½ ( x + 2 ) = 1

[Wayne]

Solve and show working.
Slightly harder.

½ ( x + 2 ) = 1

1/2(x+2)=1
x+2=1x2=2 (change side, division become multiplication)
x=2-2=0 (change side change sign)

[McDuck]

1/2(x+2)=1
x+2=1x2=2 (change side, division become multiplication)
x=2-2=0 (change side change sign)

yup,

½ ( x + 2 ) = 1

Multiply both sides by 2 to get rid of the half .

½ . 2 ( X + 2 ) = 1. 2

or

x + 2 = 2

Subtract 2 from both sides.

x + 2 - 2 = 2 - 2

so

x + 0 = 0

or

x = 0

[McDuck]

Let's play on the other side of nothingness.


20 + ( - 45 ) + 10 + ( - 1) = ?

[Wayne]

Let's play on the other side of nothingness.


20 + ( - 45 ) + 10 + ( - 1) = ?

=20-45+10-1 (removing brackets because there is nothing to calculate in there)
=-16

[McDuck]

Abstract question.



How can you have -16 of something?

You have 5 jellybeans, how can you eat 7 jellybeans and end up with - 2 jellybeans.

How can you have -2 of anything.

Surely its preposterous.

We expect the rules of Math to be reliable and accurately reflect the real world.

What's really going on?