Wednesday, April 29, 2015

448: Hexagon Puzzle

Why did the puzzle creator only ask for circles A and B?

In the diagram, the 19 circles are to contain the integers 1 to 19 once each.
The sum total along any of the lines is 38.
Determine the integers in circles A and B.

Can you find the rest of the circles, too?


Another puzzle from Five Triangles.

Tuesday, April 28, 2015

447: Shading a Square


Q1: How many different ways are there to shade this to show ¼?
Q2: How might kids shade this to show ¼?

Source:

Sunday, April 26, 2015

445: Early Hard Drives


This 1961-era 1MB hard drive is too small to hold this picture.

This 28MB IBM 1301 machine cost $110,000 and weighed half a ton.



And now you have this to carry around in your pocket.


Go here:
http://www.jcmit.com/flashprice.htm

Can you compose a simple rule about prices over time?

Data copied here:

X Y Year Month Sales Company Manufacturer Size Size Cost
date US$/MB           MBytes US$
2003.17 0.258 2003 25-Mar NewEgg.com Samsung 128MB Flash 128 33
2003.58 0.191 2003 19-Aug NewEgg.com Dana-Elec 256MB Flash 256 49
2003.75 0.219 2003 1-Oct NewEgg.com Delkin 256MB Flash 256 56
2004 0.191 2004 Jan pcclub.com Simpletech 256MB Flash 256 49
2004.17 0.238 2004 16-Mar NewEgg.com SanDisk 256MB Flash 256 61
2004.33 0.266 2004 4-May NewEgg.com SanDisk 256MB Flash 256 68
2004.42 0.223 2004 8-Jun NewEgg.com SanDisk 256MB Flash 256 57
2005.5 0.0762 2005 July Crucial.com Crucial 512MB Flash 512 38.99
2006.33 0.0219 2006 25-May NewEgg.com Transcend 2GB Flash 2048 43.98
2006.67 0.0156 2006 1-Sep NewEgg.com Patriot 4GB Flash 4096 63.99
2006.83 0.0146 2006 25-Nov NewEgg.com PNY 2GB Flash 2048 29.98
2007 0.0122 2007 28-Jan NewEgg.com Wintec 4GB Flash 4096 49.98
2007.17 0.0104 2007 15-Mar NewEgg.com Transcend 8GB Flash 8192 84.98
2007.33 0.0105 2007 7-May NewEgg.com Kingston 2GB Flash 2048 21.58
2007.5 0.00976 2007 9-Jul NewEgg.com R-Data 4GB Flash 4096 39.98
2007.67 0.00805 2007 30-Sep NewEgg.com A-Data 4GB Flash 4096 32.98
2007.75 0.00763 2007 16-Oct NewEgg.com A-Data 16GB Flash 16384 124.98
2007.83 0.00732 2007 10-Nov NewEgg.com Corsair 8GB Flash 4096 29.99
2007.92 0.00763 2007 5-Dec NewEgg.com Corsair 16GB Flash 16384 124.99
2008 0.00458 2008 18-Jan NewEgg.com A-Data 16GB Flash 16384 74.98
2008.08 0.00463 2008 20-Feb NewEgg.com PQi 4GB Flash 4096 18.98
2008.33 0.00415 2008 18-May NewEgg.com A-Data 16GB Flash 16384 67.98
2008.5 0.00354 2008 7-Jul NewEgg.com A-Data 16GB Flash 16384 57.98
2008.58 0.00305 2008 21-Aug NewEgg.com A-Data 16GB Flash 16384 49.98
2008.67 0.00311 2008 25-Sep NewEgg.com Transcend 16GB Flash 16384 50.98
2008.83 0.00183 2008 14-Nov NewEgg.com A-Data 16GB Flash 32768 59.99
2008.92 0.00214 2008 14-Dec NewEgg.com Corsair 32GB Flash 32768 69.99
2009 0.00162 2009 19-Jan NewEgg.com OCZ 16GB Flash 16384 26.49
2009.08 0.00168 2009 14-Feb NewEgg.com HP 8GB Flash 8192 13.75
2009.25 0.00204 2009 3-Apr NewEgg.com Patriot 64GB Flash 65536 133.99
2009.42 0.00214 2009 3-Jun NewEgg.com A-Data 16GB Flash 16384 34.99
2009.5 0.00195 2009 2-Jul NewEgg.com A-Data 16GB Flash 16384 31.99
2009.58 0.00177 2009 20-Aug NewEgg.com A-Data 32GB Flash 32768 57.99
2009.75 0.00214 2009 7-Oct NewEgg.com Corsair 16GB Flash 16384 34.99
2009.92 0.00226 2009 12-Dec NewEgg.com A-Data 16GB Flash 16384 36.99
2010 0.00244 2010 15-Jan NewEgg.com Corsair 32GB Flash 32768 79.99
2010.08 0.00226 2010 11-Feb NewEgg.com Lexar 16GB Flash 16384 36.99
2010.17 0.00244 2010 10-Mar NewEgg.com HP 8GB Flash 8192 19.99
2010.33 0.00201 2010 9-May NewEgg.com PNY 16GB Flash 16384 32.99
2010.5 0.00193 2010 25-Jul NewEgg.com Kingston 8GB Flash 8192 15.85
2010.58 0.00176 2010 26-Aug NewEgg.com Lexar 16GB Flash 16384 28.85
2010.75 0.00177 2010 7-Oct NewEgg.com Kingston 16GB Flash 16384 28.99
2010.83 0.00146 2010 16-Nov NewEgg.com OCZ 16GB Flash 16384 23.99
2010.92 0.00119 2010 10-Dec NewEgg.com Patriot 32GB Flash 32768 38.99
2011 0.001403 2011 21-Jan NewEgg.com Transcend 16GB Flash 16384 22.99
2011.08 0.001251 2011 12-Feb NewEgg.com Verbatim 32GB Flash 32768 40.98
2011.33 0.001342 2011 6-May NewEgg.com Patriot 16GB Flash 16384 21.99
2011.42 0.001219 2011 22-Jun NewEgg.com Transcend 16GB Flash 16384 19.98
2011.67 0.001007 2011 2-Sep NewEgg.com Team 32GB Flash 32768 32.99
2011.75 0.001037 2011 17-Oct NewEgg.com Verbatim 16GB Flash 16384 16.99
2012 0.000824 2012 12-Jan NewEgg.com Corsair 32GB Flash 32768 26.99
2012.08 0.000824 2012 13-Feb NewEgg.com Team 32GB Flash 32768 26.99
2012.25 0.000793 2012 8-Apr NewEgg.com Team 32GB Flash 32768 25.99
2012.33 0.000702 2012 24-May NewEgg.com HP 32GB Flash 32768 22.99
2012.58 0.00058 2012 20-Aug NewEgg.com Transcend 32GB Flash 32768 18.99
2012.62 0.00061 2012 21-Sep NewEgg.com Kingston 16GB Flash 16384 9.99
2012.83 0.00061 2012 17-Nov NewEgg.com Kingston 16GB Flash 16384 9.99
2013 0.000458 2013 17-Jan NewEgg.com DNY 64GB Flash 65536 26.99
2013.08 0.000518 2013 23-Feb NewEgg.com Lexar 32GB Flash 32768 16.99
2013.33 0.00058 2013 18-May NewEgg.com Transcend 64GB Flash 65536 37.98
2013.42 0.000564 2013 30-Jun NewEgg.com Transcend 64GB Flash 65536 36.98
2013.58 0.000564 2013 9-Aug NewEgg.com Transcend 64GB Flash 65536 36.98
2013.62 0.000564 2013 7-Sep NewEgg.com Transcend 64GB Flash 65536 36.98
2013.75 0.000534 2013 16-Oct NewEgg.com Centon 64GB Flash 65536 34.98
2013.83 0.000542 2013 10-Nov NewEgg.com Centon 128GB Flash 131072 70.98
2013.92 0.000548 2013 23-Dec NewEgg.com AData 16GB Flash 16384 8.98
2014.08 0.000457 2014 8-Feb NewEgg.com Transcend 32GB Flash 32768 14.98
2014.17 0.000473 2014 13-Mar NewEgg.com Team 64GB Flash 65536 30.98
2014.25 0.000366 2014 10-Apr NewEgg.com Silicon Power 32GB Flash 32768 11.99
2014.42 0.000389 2014 19-Jun NewEgg.com PNY 128GB 131072 50.98
2014.58 0.000335 2014 2-Aug NewEgg.com Adata 32GB Flash 32768 10.98
2014.67 0.000345 2014 13-Sep NewEgg.com Silicon Power 64GB Flash 65536 22.58

Saturday, April 25, 2015

444: Floorlamp

The floor lamp casts a shadow.
Straight line AB is drawn on a wide, spacious floor. A lamp is at a height of 8  m above point C, which is located on line AB.  Line CD is perpendicular to line AB.  A rectangular solid 5 m by 3 m by 4 m is on the floor as shown.  The box is 2  m from line AB and a meters from line CD. Assume the lamp is a point source and will cast a perfect shadow. Given that the rectangular solid casts a shadow with area 90  m², determine the distance a (in meters).

  • Does the 2m distance from the wall matter to the shape of the shadow?
  • How does the distance a change the area of the shadow?
  • What shape is that shadow on the floor? 
  • Are the edges of the shadow guaranteed to parallel with the edges of the box?
  • How far away is that box from the light? 
It's another Five Triangles puzzle

Thursday, April 23, 2015

442: Square Root

x² = 81 has two solutions, -9 and 9.

But does √81 have one solution or two? Is it correct to say that √81 = +9 and -9?

Or should we be saying that √81is an expression and that 9, 18/2, 27^(2/3), and 1+6+2, are equivalent expressions?

Wednesday, April 22, 2015

441: How many K?

Edited out the answer and some hashtags  ...

For how many integers k is 10,000 - k² positive?

Tuesday, April 21, 2015

440: Revenge of the Mad Hatter

There are three red hats and two black hats on a hatstand.

Melba, Fran, and Brent line up in a single file line, and the mad Hatter places a hat on each person’s head and tosses the rest into a box where no one can see them.

The Mad Hatter takes off Brent's blindfold. Brent, who could see Fran’s hat and Melba’s hat, says, “I don’t know what color my hat is.”

The Mad Hatter takes of Fran's blindfold. Fran, who could only see Melba’s hat, says, “I heard what Brent said, but I don’t know what color my hat is.”

As the Mad Hatter moved to take off Melba's hat, she said, “I don't need you to take off my blindfold. I know what color my hat is.”

What color is Melba’s hat and how does she know?

Monday, April 20, 2015

439: Fraction / Decimal Tic-Tac-Toe

We all know that playing the middle of a tic-tac-toe game ensures that you can never lose if you play it right, but this one seems to be a bit trickier.

Where is the best place to start?


Sunday, April 19, 2015

438: NY Daily Statistics

NY Daily News had this:
Government-hating gun owners are shooting themselves in the foot.

The gun-buying frenzy triggered by the 2012 Sandy Hook school shooting pumped $875 million into federal coffers in 2013 alone, according to Vocativ.com. That’s because the Treasury’s Alcohol and Tobacco Tax and Trade Bureau collects a 10% tax on ammo sales and a nearly 10% tax on gun sales. The tax revenue on guns and ammo reached an all-time high the year after the Sandy Hook Elementary School massacre, which left 27 students and teachers dead and triggered fears of stricter gun laws.

The roughly $255 million in tax revenue on ammunition alone brought in by the feds in 2013 was nearly double the 1991 tax receipts for handguns, long guns and ammo combined, according to Vocativ.

Is that graph properly done?
Is it misleading?
Is the data itself misleading?
source.

Saturday, April 18, 2015

437: Specificity and Clarity in Math

Mathematical Grammar ... I think a comma might be necessary in #2, maybe two of them. What do you think, students?

436: Whiteboards at NCTMBoston


435: Interesting Question from NCTM Boston


Friday, April 17, 2015

434: Integer Quotient

Is it possible to find a positive integer value of  n so that this quotient is an integer?

Thursday, April 16, 2015

433: Do We Need to Feed the Meter?






If you are parked during any part of the time indicated, then you need to pay, even if it's only for a few minutes at the beginning or end of your shopping. (If you only are there for the five minutes between 8:00 and 8:05, you only pay a pro-rated fee.)


If you park at any random time and leave your car there for an hour, are you more likely to pay or more likely to not pay?


source.

Wednesday, April 15, 2015

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.

Tuesday, April 14, 2015

431: Mr Smith has Two Children.

Thank you for reminding me of this:

Monday, April 13, 2015

430: Dangerous Mistake

What is the more critical mistake, the more damaging to civilization as we know it?

"Me and John are going to the mall" 
or
"1/2 + 1/3 = 1/5"

Sunday, April 12, 2015

Saturday, April 11, 2015

428: Surface area of that cube again.


Let's use this Rubik's Cube again, shall we? The total surface area of the cube is 54 square units.
  • Is it possible to remove a unit cube without changing the surface area?
  • How about removing one that increases the area by 2 square units ....
  • or maybe one that can be removed that increases area by 4 sq units?

Friday, April 10, 2015

427: More water in Boxes


A closed rectangular box whose dimensions are 8 feet by 5 feet by 3 feet has some water in it. When the box is resting on its smallest face, the water is 5 feet deep. How deep will the water be if the box is resting on its largest face?

source.

Thursday, April 9, 2015

426: Pouring Water

Here are six sets of containers. The water in the top one is draining into the bottom one at a constant rate. The puzzle here is to determine, at five intervals during the pour, what the surface of the water looks like when viewed from above.




Here are the possible choices. There is one set for the surface of the upper one and one set for the surface of the lower one. Some of these cards have missing pieces. Students need to draw those.





Print them from the source below.


source.

Wednesday, April 8, 2015

425: Filling Bottles


Imagine you were filling each bottle with water, poured at a constant rate.

Which bottle fits which height-time graph.
Draw the bottles for the remaining graphs.

source.

And don't forget https://teacher.desmos.com/waterline 

Tuesday, April 7, 2015

424: Point-point-slope

Challenges

  1. Use four different digits (from 2-9) to create two points which determine a line with the greatest possible slope.
  2. Use four different digits (from 2-9) to create two points which determine a line with the least possible slope.
  3. Use four different digits (from 2-9) to create two points which determine a line with a slope as close to zero as possible.

source.

What does "greatest" slope mean?
What does "least" slope mean ... most negative?

Monday, April 6, 2015

423: If only

There's a neat game you can play with the English language:

Place the word "only" anywhere in the sentence. How does it change the meaning?

Only she told him that she loved him.
She only told him that she loved him.
She told only him that she loved him.
She told him only that she loved him.
She told him that only she loved him.
She told him that she only loved him.
She told him that she loved only him.
She told him that she loved him only.

What would it look like if we were trying to create a math problem that worked the same way, except "only" is replaced by the equals sign?

Sunday, April 5, 2015

422: Inscibed Semicircle, part three

Last question with this visual:

How could you draw the inscribed semicircle (area = π) so that the rectangle is of maximum size?


Saturday, April 4, 2015

421: Inscribed Semicircle, part two

This problem was posed on Twitter the other day.
A semicircle is inscribed in a rectangle. If the area of the semicircle is π, what's the area of the rectangle? [4]
My question yesterday was ... how could you draw the inscribed semicircle in a way that gives a rectangle of a very different (and larger) area?

My question today is, given the arrangement below, what points did I choose that had integer coordinates? I chose a larger semicircle - for convenience - how big was it?


Friday, April 3, 2015

420: Inscribed Semicircle

This problem was posed on Twitter the other day.
A semicircle is inscribed in a rectangle. If the area of the semicircle is π, what's the area of the rectangle? [4]
My question is ... how could you draw the inscribed semicircle in a way that gives a rectangle of a very different (and larger) area?