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.

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:

Q1: How many different ways are there to shade this to show ¼? Q2: How might kids shade this to show ¼?  pic.twitter.com/vntk6kcYw7
— David Wees (@davidwees) April 25, 2015

446: ASN - Geometry

Always, Sometimes, Never?

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

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

443: What are the questions?

source, of course.

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?

It is really alarming how often I see materials written by math teachers claiming that for example sqrt(81) is both 9 and -9.
— Kate Nowak (@k8nowak) April 22, 2015

441: How many K?

Edited out the answer and some hashtags  ...

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

For how many integers k is 10000-k^2 positive? #SATPrep #mathchat  @MathCurmudgeon
— David Marain (@dmarain) April 17, 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?

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?

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.

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?

This activity, which asks kids to connect representations, was designed by @MaeghanMath #NCTMBoston pic.twitter.com/oatBMTqTLH
— David Wees (@davidwees) April 16, 2015

436: Whiteboards at NCTMBoston

Found this in the #NCTMBoston exhibit hall. What questions do you have? #mathchat
— David Wees (@davidwees) April 17, 2015

435: Interesting Question from NCTM Boston

Interesting question. #NCTMBoston pic.twitter.com/Jco11RaesK
— Steve Wyborney (@SteveWyborney) April 18, 2015

434: Integer Quotient

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

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.

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.

431: Mr Smith has Two Children.

Thank you for reminding me of this:

Mr Smith has two children. One of them is a boy. What is the probability the other child is also a boy?
— solve my maths (@solvemymaths) April 4, 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"

429: Least Possible Value of y

Is there enough information to determine the least possible value of y?

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?
  

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.

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.