## Monday, June 30, 2014

### 172: Fleur-de-lis

"That's cool. Where's it from?"

"The Old Mississippi River bridge. There were once probably tens of thousands of these on that bridge. Both sides of each rail on each side of the road every 24" for a mile. I've got five of them."

## Sunday, June 29, 2014

### 171: Steering Wheel.

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

## Saturday, June 28, 2014

### 170: Proof Without Words 4

Can you interpret this Proof Without Words?

## Friday, June 27, 2014

### 169: Quick Fractions

Explain whether 4/3 or 3/4 is closer to 1, and how you know.

## Thursday, June 26, 2014

### 168: Knight's Tour

An engineer from New Zealand, Mr. Sturmer, published this Knight's Tour in 1888. Can you follow the tour and recreate the poem? (Mouseover the image for a hint)

## Wednesday, June 25, 2014

### 167: Probability, p5

Snakes on a plane:
When you're flying, always take a pet snake with you in your hand luggage. The probability of there being TWO snakes on the plane is almost zero, so you will be safe from snake attack.

Staying dry at the cricket match:
Follow the example of the famous mathematician Hardy and take an umbrella with you to cricket matches. If you forget your umbrella it is more likely to rain, so if you remember to take it with you it is more likely to be sunny all day.

## Tuesday, June 24, 2014

### 166: Lottery, p4

Here's some more advice for gamblers. Do you agree with it or not?

Coin Flipping:
If tails has come up on the last 9 occasions then it's a good idea to call tails again.

Winning at Roulette:
If red has come up lots of times in a row, you should bet on black next.

## Monday, June 23, 2014

### 165: Lottery, p3

In the UK, the lottery consists of picking 6 numbers between 1 and 49.  Any player to match all 6 numbers is the grand prize winner.As we all know, there's lots of free advice on how to win the lottery, usually given out by someone who, strangely, seems more willing to sell the information to you instead of using it themselves.  Odd, that. Anyway, here's some advice you just bought for a small fee:
Roughly equal numbers of odd and even are drawn most weeks, so you should pick a good mixture of odds and evens.

Never choose six numbers all from the same group - for example, all single digits, all multiples of five, all with the same last digit ...

Always pick some higher numbers from the 30s and 40s.

What do you think? Advice worth paying for?

## Sunday, June 22, 2014

### 164: Lottery, Part 2

In the UK, the lottery consists of picking 6 numbers between 1 and 49.  Any player to match all 6 numbers is the grand prize winner.As we all know, there's lots of free advice on how to win the lottery, usually given out by someone who, strangely, seems more willing to sell the information to you instead of using it themselves.  Odd, that. Anyway, here's some advice you just bought for a small fee:
When picking lottery numbers, choose numbers that sum between 100 and 200 because the total is rarely outside this range.

What do you think? Advice worth paying for?

### 163: Playing the Lottery, part 1

In the UK, the lottery consists of picking 6 numbers between 1 and 49.  Any player to match all 6 numbers is the grand prize winner.  The chances of this are certainly astronomically low.
How low?
If we bought a lottery ticket for every different combination of 6 numbers to ensure we’d win, how high would that stack of tickets reach?
How long would it take for us to purchase them?

## Saturday, June 21, 2014

### 162: Mosiac, p4

You successfully transmitted the sizes of the various figures and your correspondent has successfully created them.  For a bit of fun, you say only that the final figure will be "a circumscribed polygon with more than two lines of symmetry."

How many different ways can this be accomplished ... or is that description sufficient to repeat the figure?

Image from Dan Meyer

## Friday, June 20, 2014

### 161: Mosaic, p3

What if your description started with the circle?
• Eight Squares, side length X
• Eight parallelograms, side lengths Y (and Z?) and angles of A and B

What would those measurements be?

Image from Dan Meyer

## Thursday, June 19, 2014

### 160: Mosaic, p2

Pretend for a moment that part of your description was
• Eight Squares, side length 1
• Eight parallelograms, side lengths Y (and Z?) and angles of A and B
• and One circle, radius X.

What would those measurements be?

Image from Dan Meyer

### 159: Mosaic, p1

There's the task. Print this out for half the class (A) and have them describe to the other half (B), who draws it using the tools chosen by the descriptor (A).

paper, pencil, compass, straightedge?
Geogebra?
Tangram pieces?

Can they use their cellphones and call their friend?

From Dan Meyer

## Wednesday, June 18, 2014

### 158: Equations with Symmetry

Find four distinct integers a, b, c, and d such that ab = c + d and cd = a + b.

From NCTM.

## Tuesday, June 17, 2014

### 157: Angles

Let's change this slightly ...

With a spinner pointer starting at 0 degrees (positive x-axis), rotate by every INTEGER angle between 0° and 360°. Where do you end up?

## Monday, June 16, 2014

### 156: Cheating?

Is this evidence of cheating?

## Sunday, June 15, 2014

### 155: Pre-Medieval Math, Pyramids part two

Here are the pyramids at Giza, viewed from a satellite. Notice that they are aligned to the north-south lines ... in fact, to within a 0.05 degree. Also measurements of the plateau they sit on have determined that the foundation was carved into the bedrock and the entire site was leveled to within a fraction of an inch.

It's known how the Egyptians managed to perform this feat with the simple tools they had ready and now it's time for you to consider it and try to figure it out..

String, standard measuring sticks carved from granite, a good pair of eyes, that's all. Remember, they were just as smart as you and you are just as smart as they.
1. How did they align the edge N-S?
2. How did they level the base so precisely?
Give up? Look at the alt text of the photo.

Was it aliens?

No, that's just crazy talk.

Look at the alt text for the pyramid photo for the explanation.

Here's another ..

## Saturday, June 14, 2014

### 154: Pre-Medieval Math, Pyramids

From Bryn, O.J. 2010. Retracing Khufu’s Great Pyramid, Nordic Journal of Architectural Research vol. 22, no 1/2, 135-144.

You must enlarge this to see the detail.

Interpret the construction layout, Find the angle of the sides .... heck, you know how this works.

## Friday, June 13, 2014

### 153: More Manuscript Geometry

Straightedge, compass and a pencil ... GO.

## Thursday, June 12, 2014

### 152: Medieval Geometric Constructions

Ok, you have a straightedge and a compass, and a steel point to scribe with.
Geometrical constructions ....
GO.

Symmetry.

Images are of the Alhambra, of course.

## Wednesday, June 11, 2014

### 151: Medieval Altitude Measurement

Explain the reasoning used in 1588 to find the height of the castle.

So they could shoot it.

## Tuesday, June 10, 2014

### 150: Angle of Elevation in 1588

If that inclinometer says 40° between the cannon's mouth and plumb, what's the angle of elevation?

## Monday, June 9, 2014

### 149: Medieval Ballistics

Tartaglia

Using the angle between line of sight to the projectile and the horizontal, show why the trajectory was considered to be in this shape for many centuries. (Hint: consider that only those viewing from BEHIND could see the projectile. UPDATE: Yes, those in front could see something coming. "I couldn't figure out what it was coming closer and closer ... then it hit me.")

This may be more than you want ... if so, pay no attention to the following instructions.

In Geogebra, graph the parabolic arc. We'll assume for simplicity that the cannonball begins at (0,0) and hits at (1000,0) with max at (500,250). In the input box, place y=x(x-1000)/1000. Place A at (-10,0) and B, a point on the object f(x). Place c(500,0) so we have three points. Create line segments AB and AC. Measure CAB. Drag B along the curve, paying attention to the angle.  What do you see?
How does it change as you move B?

## Sunday, June 8, 2014

### 148: Medieval Math - Polyhedra

From manuscript dated 1585,
Houghton Library. MS Typ 108. Geometrica et Perspectiva Corpora Regulata et irregulata.

## Saturday, June 7, 2014

### 147: Exponents

If a, b are positive integers greater than 1 and ba=212 then what is the largest possible value of b?

from D. Marain

## Friday, June 6, 2014

### 146: Cavalieri's Principle

At \$7.50 per square meter, how much is this walkway going to cost?

## Thursday, June 5, 2014

### 145: Proof Without Words 3

Can you interpret this Proof Without Words that someone carelessly threw in the trash?

## Wednesday, June 4, 2014

### 144: Proof Without Words 2

Can you interpret this Proof Without Words?

## Tuesday, June 3, 2014

### 143: Proof Without Words 1

Can you interpret this Proof Without Words?

## Monday, June 2, 2014

### 142: Price of HotWings

What do you think was the most popular option?

A restaurant I recently visited had the following options on their menu:
• 10 wings for \$7.99 with two sauces
• 15 wings for \$12.49 with two sauces
• 20 wings for \$16.49 with two sauces
• 30 wings for \$24.79 with three sauces
• 50 wings for \$39.79 with four sauces

Source.

## Sunday, June 1, 2014

### 141: Topology Bagels - One Ring

Yesterday's cutting was interesting, but difficult to achieve. This one is simpler. Can you cut your bagel so that you get one continuous loop of bread?