Why Social Media Might Make Companies Less Generous – Game Theory Tuesdays

Posted October 18, 2016 By Presh Talwalkar

image by frankieleon. CC BY 2.0
image by frankieleon. CC BY 2.0

Years ago I bought a 12-pack of Coca Cola and stored it in my pantry. The next day I saw one can had a small pinhole puncture that dried up into a syrupy mess in the cardboard box packaging. After I called, customer service apologized and promptly mailed a physical coupon for free Coke.

At that time, companies often were generous to complaining customers: cell phone companies gave free minutes, magazines and newspapers extended subscriptions, and companies mailed coupons. Occasionally you could even get free stuff–without even complaining–merely by asking. One person spent about $50 to write letters to 35 companies just asking for free stuff. The result is he got lots of free stuff, like free White Castle hamburgers, four Sanford pens, a BMW keychain, and various other samples and coupons, totaling over $270.

Others have reported similar success: see The Thrillist and ToughNickel.

You might think companies would be even nicer today because social media means their outstanding service would be shared and spread around. But if companies know they are being watched, they might actually be less likely to give out freebies. Let’s consider why.
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Can You Solve This Geometry Problem For Singapore Students? Finding The Area Of A Claw. Sunday Puzzle

Posted October 16, 2016 By Presh Talwalkar

This geometry problem was asked to students in Singapore. What is the area of the shaded region? The figure is composed of squares with side length 4 and quarter circles.

singapore-geomety-problem-quarter-circles-claw

Watch the video for a solution.

Can You Solve This Geometry Problem From Singapore? The Area Of A Claw

Or keep reading.
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The Wallis Product Formula For Pi And Its Proof

Posted October 12, 2016 By Presh Talwalkar

Start with the fractions 2/1 and 2/3.

(2/1)(2/3)

Increase each number by 2 to get a new pair of fractions 4/3 and 4/5, and multiply this pair with the previous fractions:

(2/1)(2/3)(4/3)(4/5)

Now increase the last pair of fractions again, and add this pair to the product:

(2/1)(2/3)(4/3)(4/5)(6/5)(6/7)

Each new pair is [2n/(2n – 1)][2n/(2n + 1)]. Repeat this indefinitely and multiply all terms together.

wallis-product-formula-proof

The product, as n goes to infinity, is known as the Wallis product, and it is amazingly equal to π/2 ≈ 1.571.

The beginnings of the formula come from work in 1655. In the following video I explain a bit of how it was found historically and then I give a modern proof using calculus.

The Wallis Formula For Pi And Its Proof

Or keep reading.
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SciShow Incorrectly Explains The Nash Equilibrium. Game Theory Tuesdays

Posted October 11, 2016 By Presh Talwalkar

SciShow is one of the most popular education channels on YouTube with 3.7 million subscribers. A recent video is a about game theory, which has over 300,000 views.

Game Theory: The Science of Decision-Making

The video correctly explains a number of concepts. However, there is an egregious error in its explanation of the Nash equilibrium. Can you figure out the mistake? Here is the pertinent slide, from 3:30 in the video.

sci-show-nash
(Image from YouTube)

Video text: “A player in a game has found Nash Equilibrium when they make the choice that leaves them better off no matter what their opponents decide to do.”

This is a fundamental misunderstanding of the definition that would merit failing a game theory class. I explain the mistake below.
Continue reading SciShow Incorrectly Explains The Nash Equilibrium. Game Theory Tuesdays

The Mind-Twisting Logical List Riddle – Sunday Puzzle

Posted October 9, 2016 By Presh Talwalkar

Which of the statements in this list of 10 are false? The line n states there are exactly n false statements in the list.

#1. Exactly 1 of the statements in this list is false.
#2. Exactly 2 of the statements in this list are false.
#3. Exactly 3 of the statements in this list are false.
#4. Exactly 4 of the statements in this list are false.
#5. Exactly 5 of the statements in this list are false.
#6. Exactly 6 of the statements in this list are false.
#7. Exactly 7 of the statements in this list are false.
#8. Exactly 8 of the statements in this list are false.
#9. Exactly 9 of the statements in this list are false.
#10. Exactly 10 of the statements in this list are false.

How many are false, if line n reads that “at least” n items are false?

#1. At least 1 of the statements in this list is false.
#2. At least 2 of the statements in this list are false.
#3. At least 3 of the statements in this list are false.
#4. At least 4 of the statements in this list are false.
#5. At least 5 of the statements in this list are false.
#6. At least 6 of the statements in this list are false.
#7. At least 7 of the statements in this list are false.
#8. At least 8 of the statements in this list are false.
#9. At least 9 of the statements in this list are false.
#10. At least 10 of the statements in this list are false.

Watch the video for a solution.

The Mind-Twisting Logical List Riddle

Continue reading The Mind-Twisting Logical List Riddle – Sunday Puzzle