Spiked Math

Classroom Anatomy

2012-05-18T03:24:11Z

SpikedMath

Summer Break

2012-05-01T18:23:01Z

It's May 1st so what that means is Spiked Math will now operate on the Summer schedule.

May 1 -- August 31 "summer" schedule:

Aiming for updates at least once a week (say on Tuesday or Wednesday)
Sporadic updates on other days

Textbook

2012-04-30T23:12:49Z

SpikedMath

The Fundamental Theorem of Matrix Theory

2012-04-30T01:45:43Z

This constructive proof works for any m x n matrix!

Relevant xkcd: http://xkcd.com/184/

Tired of adding C?

2012-04-29T03:02:29Z

Warning: You may lose marks depending on the teacher!

Epic Arrow / Epic Morphism

2012-04-28T03:58:40Z

Now that everyone has had a chance to catch up on their Category Theory, you should have no problem with today's comic!

Or if you don't use the double headed arrow notation, you might prefer the following:

Hint: http://en.wikipedia.org/wiki/Epimorphism

Math Jeopardy

2012-04-26T19:51:28Z

Come at me Ken Jennings, the ball is in my court now!

Hint: The categories are Top, Grp, Set, Vect_R, Rng.

Relevant Link: http://en.wikipedia.org/wiki/Category_theory

Math Jokes

2012-04-25T18:21:38Z

Link: http://jcdverha.home.xs4all.nl/scijokes/1_6.html

And a further correction as quoted by Nils B.:
"You've perhaps heard from 17,000 mathematicians since this section was quoted in the 2005 Jan "Notices of the AMS", but while splitting hairs, this definition of compact is incorrect: it defines -totally bounded-, not compact. For instance, the -open- interval (0,1) is totally bounded (can be guarded by a finite number of policemen, no matter how nearsighted a policeman is) but not compact."

Scumbag Gauss

2012-04-25T03:51:40Z

Gauss quite often did not publish his findings setting mathematics back by 50 years! (according to mathematical historian Eric Bell).

Also, Scumbag Mike for making meme comics.

lolcatenary

2012-04-24T03:57:10Z

Relevant link: http://en.wikipedia.org/wiki/Catenary

And a bonus comic!!!

Relevant Link: http://en.wikipedia.org/wiki/Pascal%27s_triangle

Should I cite this paper?

2012-04-20T15:05:58Z

First a "comic":

And now a puzzle - Can anyone beat hardcore? And HOW!!!

Curling with gradients

2012-04-19T02:45:17Z

I did a google search for "gradients can't curl" and came up with Qiaochu Yuan's comment, so credit to him!

Epsilon-delta comics

2012-04-18T07:06:05Z

Credit to Ken S. for submitting this comic idea:

Relevant Link: http://mathworld.wolfram.com/Epsilon-DeltaProof.html

Is math beautiful?

2012-04-13T01:57:17Z

So, is math beautiful?

http://en.wikipedia.org/wiki/Borwein_integral

MFT - Lake Escape

2012-04-11T03:15:47Z

Math Fact Tuesday! More of a puzzle, but if I give you the answer it'll become a fact, right??

This puzzle (credited to Martin Gardner) can be stated as follows (text source):

We are in a rowboat in the center of a circular lake. On the edge of the lake is a mean goblin who wants to eat us; and if he catches us, he will do so!! The goblin can't swim and won't go into the lake (and doesn't have a boat), but he can run K times as fast as we can row.

We, however, can run significantly faster than the goblin can, so if we are able to reach a point at the edge of the lake without the goblin being there, then we will be able to escape.

Question 1a: For K = 1, is there a strategy we can use to escape?

Question 1b: What about K = 3? or K = pi? or K = 4? or K = 5?

Question 2: What is the best possible value of T so that:

if K>T then the goblin has a strategy to catch you (and eat you),
if K<=T then we have a strategy that allows us to escape from the lake without being eaten.

A very nice demonstration (game) due to Alexey Izvalov is below. The levels are as follows:

Level 1 - Goblin's speed is 3 times faster than your rowing speed (K = 3).
Level 2 - Goblin's speed is 3.25 times faster than your rowing speed (K = 3.25).
Level 3 - Goblin's speed is 3.5 times faster than your rowing speed (K = 3.5).
Level 4 - Goblin's speed is 3.75 times faster than your rowing speed (K = 3.75).
Level 5 - Goblin's speed is 4 times faster than your rowing speed (K = 4).

If you think the answer in Question 2 satisfies T>=4, then you should be able to beat level 5! Otherwise, good luck trying!

Now for a solution.

One (optimal) strategy that works for the rower is to trace out a letter J (under the assumption that the goblin does not change direction). If the goblin changes direction at some point, you can modify your strategy to do even better. Rather than rewriting out the entire solution for this strategy I'll just refer you to this link: http://domino.watson.ibm.com/Comm/wwwr_ponder.nsf/solutions/May2001.html

Then the answer to Question 2 (after a bit of work) is the solution T to the pair of equations:
cos(B) = 1/T,
sin(B) = (1/T)*(pi + B).
Solving this gives: T=4.6033388...

What this means is that you can win the game above on level 5! (and if there were a level 6 with K=4.6, you can likely win with some careful mouse movements).

If you would like to see exactly how an optimal solution works for K=4.6033388... then visit the link (http://openmap.bbn.com/~tomlinso/chaser/chaser.html) and view the simulation applet on the right.

Lower case xi?

2012-04-09T16:41:47Z

Fun fact: My xi looks exactly the same as my right curly bracket } (and coincidentally, my left curly bracket too!).

BONUS: My first experience with xi: http://img.spikedmath.com/comics/498-xi2.png

MPF - Grid Game

2012-04-07T01:08:43Z

Question 2: What if Randall goes first?

Question 3: What if it is an m x n grid?

Tons of great genie responses from yesterday's comic and even an illustration in the Physics Forums! It's gonna be tough to choose a winner, but on the other hand, I do have a lot of creepy hugs to go around.

The Math Genie

2012-04-04T21:00:51Z

It's your turn now!

1. Name a wish.
2. Describe how the "Math Genie" will grant that wish!

Who ever comes up with the most clever Math Genie wish & response will win an email from Mike saying "Congrats!" (Note: The email may be traded in for one creepy hug instead).

Gabriel's Horn

2012-04-03T03:27:19Z

You've been warned.
http://en.wikipedia.org/wiki/Gabriel%27s_Horn

Courtney posted a new Brown Sharpie comic!!

Thanks Veronique for the xkcd link which is most likely the motivation for the comic! I totally forgot about that comic and didn't make the connection when I posted this (but who knows what goes on in Mike's head).

Hexadecimal Grading

2012-03-29T18:40:02Z

http://en.wikipedia.org/wiki/Hexadecimal