Sunday, May 31, 2015

475: Circle Theorems

Here are eight circle theorems, written by a Brit so you'll have to "translate" to American English ...
Are these the most important ones?
Are there any that are missing that you feel should be included in this group?


474: Fractional Understanding

What insights does a student need to have before this problem is solvable?

Friday, May 29, 2015

473: Overlapping Squares

I might have posted this puzzle before.

What do your students think? Can they generalize it?
What is the overlapped area?

Sunday, May 24, 2015

472: How Many geometry Theorems Are There?

That's pretty much it: How many Geometry Theorems are needed to solve this extended problem?


471: Digits in Fibonacci

We all recognize the Fibonacci Sequence:  1, 1, 2, 3, 5, 8, 13, 21, ...
What is the units digit of the sixty-first Fibonacci number?

Is there a pattern?

470: Poor Alex Bellos

Normally, I'm glad when the nedia talks about math, even if it is simply the latest "Super-Hard problem from Asia that All of Us Stupid American Adults and Brits Won't be Able to Solve but Their 8 Year-old Do Routinely."

Throw in "I gave it to a mathematician and he couldn't solve it either" and you've got an instant viral hit, apparently.

All right, rant over.

What's the error?

Oh, and solve it if you want, too. It's not really that hard.

Further questions:
  • Which blanks CANNOT have a prime number in them?
  • Which blanks COULD have certain prime numbers?
  • Which blanks essentially force this problem to be non-unique?

(answer linked from there)

469: Fix this Graph #ILostCount

This graph needs a lot of work.

What information is being presented?
What is the best way to display this information?
Does this graph rely too much on the reader's knowledge of the city?


Tuesday, May 19, 2015

468: Fields of Green

What is the area of the whole hexagonal shape?

Can your students generalize this result?


Monday, May 18, 2015

467: Squares

Take a square number and multiply by four. Is it guaranteed to still be a square number?

Can we generalize this rule?
Can your students explain why?


Sunday, May 17, 2015

465: Probability Machine

You've all seen Plinko and its variations.  Here's one:

This one has a normal curve drawn on the background. Should it be a normal curve or more triangular like the arrangement of the pins?

Would it still be in this shape if the pins were arranged in a rectangle?

464: Illusion 4 - Side View

How did he pull this off?

Can you sketch this scene from the side?

463: Illusions 3 - Cafe Wall

Here's the original Cafe Wall, after which the Cafe Wall Illusion was named.

What went wrong with this wall? What is it that every single bricklayer and tile-setter knows as Rule #1, to avoid the Cafe Wall Illusion?

462: Illusions 2

How did this artist lay out the guidelines on this flat, concrete floor for this seemingly 3D artwork?

460: Illusions 1

What is the best way to show that the horizontal lines are, in fact, horizontal and parallel?

Why does your brain insist that they are not?

Thursday, May 14, 2015

459: Cubes

Sure, you can ask, "How many?"

But wouldn't you rather see if your students can figure out what kind of fruit is each cube?

Wednesday, May 13, 2015

458: Is That You, Pythagoras?

If we stipulate that $(x+12)^2+(y+4)^2+(z+3)^2=0$, then


Should we brute-force this or is there a more beautiful or subtle way of getting what we want?


Sunday, May 10, 2015

457: Codebreaker

What are the values of the letters that will allow you to go as far as possible?

How high can you go?

Is there a better way to solve this than just marching up the line, and guessing at the values for H, R, then for F, U, then for I, V and so on?

The original is below. Is this the best set of values?


456: A and B

The problem speaks for itself.

Wednesday, May 6, 2015

454: Fraction Sum

Seems like a tough problem.

"Wait a minute. That would make it easier ..."


Sunday, May 3, 2015

453: Kong Size Cereal Box

What do you think? Where's the mistake and why do you think they presented it this way?


452: Rose Petals

Go to this page at Desmos, and hit play. Explain what you see and why it is behaving that way.

There is obvious symmetry in the figure at each value of a but should we be defining something new to describe the sequence of images? A dynamic symmetry, of sorts?

Saturday, May 2, 2015

451: Growth Rate

Population of organism after n minutes is $P(n) = k*9^{(\frac{n}{2})}$.
What is the growth factor?

... algebraically stated, what is the ratio of P(t+1):P(t)?


450: One-third of a triangle

You've been asked to shade one-third of this triangle.

And how do you know?
Here's a response:

  • Is this a valid way to get one-third of the triangle?
  • Does this technique require that we stipulate a right triangle?
  • Does this technique require that we stipulate an isosceles one?
  • Do the lines have to be parallel to each other?
  • Do the lines have to be parallel to a side for this to work? 
  • Perpendicular to a side for this to work?

  • How might we generalize this method (if it can be generalized)?


449: Fractions

The sum of ¼ and a number is equal to the product of ¼ and the number.
(a) Explain why the number must be negative.
(b) What's the number ?