Inequality of the Means

Indiatimes is running an edited version of this article, which is the next installment of the puzzle column. This article is in collaboration with Prof. Krishnan Shankar of the University of Oklahoma.

The main problem in this article, the geometric version of the Inequality of the Means, has been chosen for its simplicity and elegance. It is one of those mathematical problems that is easy to state but ridiculously hard to prove. There are two puzzles this month. Please be sure to use the two different Subject lines mentioned in the article to help us distinguish which puzzle it is you are replying to.

INEQUALITY OF THE MEANS

This article is in collaboration with Prof. Krishnan Shankar, Professor of Mathematics at the University of Oklahoma.

The Oracle Asks

This article’s material came forth from the fertile mind of the extraordinary John Conway. References to the mathematics of the problem are listed at the end. It was a pleasure to discuss this problem with Prof. Dror Bar-Natan. Prof. Bar-Natan’s exposition and Javascript applet make the subject come alive on his website (http://www.math.toronto.edu/~drorbn/), which is certainly worth a visit.

We start with a well-known inequality from high school algebra: let a and b be any two non-negative numbers. Then, their arithmetic mean is at least as large as their geometric mean, i.e.,

square_root(a *b ) <= ((a + b)/2)

Equality occurs precisely when a = b. This is not hard to prove algebraically, but here is a nice geometric proof. Consider the following figure where a square of side length a + b encloses 8 right angled triangles of orthogonal sides a and b each. 

We're back, folks!

I have just started a new blog called "Ask the Delphic Oracle" as part of my work for my Times of India Group column. I won't be making very many further posts on this here blog while I am doing that, and so there will be a continued blogging hiatus over here, but while I am here today, I think I will throw out a mention to "Wonder Village", a game by Digital Green. If you have a couple of minutes to spare, please do check out Wonder Village. Digital Green is a non-profit based in India that I have been doing some work for for the past few years. It is an agricultural organization to help small and marginal farmers in India. It is currently based in New Delhi.

Anyway, here is a video of the game screen.



The game was developed by a team of just four developers and two designers in a mere five months. It is, I believe, one of the first of its kind - a real-time multiplayer simulation social network game with a development economics theme. Level up and prosper!

Ask the Delphic Oracle

Indiatimes' main page carried an edited version of this article on Operations Management and the applications of mathematical analytics there.

Ask the Delphic Oracle

This article is in collaboration with Prof. Krishnan Shankar, Professor of Mathematics at the University of Oklahoma. 

“Ask the Delphic Oracle” is a new column in the Times of India. As part of this column, we plan to run a new puzzle every month. We will allow three to four weeks for you to solve the puzzle. Please write in with your answers to: askthedelphicoracle@gmail.com. We will publish the names of the people who answered the puzzle correctly (randomly chosen out of the first fifty). Good luck!

Ask The Oracle:

Q. I am an Australian in California. I have noticed that a lot of Indians here drive Toyotas. Why do so many Indians drive Toyotas?

Answer. While we put our business analyst hats on, may we point out that there are excellent reasons to own a Toyota? The main reason is, of course, the quality of the car. But how is Toyota able to produce cars of such high quality? Behind the answer to this question lies the story of the machine that changed the world.

Before there were Hondas and Toyotas, there were Fords. The big idea that Henry Ford came up with was that of the assembly line. Henry Ford realized that if you organize a car factory floor like a meat packing assembly line where each worker gets to specialize on one piece of the job, then the productive efficiency dramatically increases. From this was born the modern automobile assembly operations setup, the machine that changed the world.  The Ford automobile assembly operations setup was further improved upon by the Toyota Motor Company by means of the Toyota Production System. The Toyota Production System consists of a unique combination of social and technical processes that makes it possible for them to create very high quality cars with low rates of failure. This makes Toyotas cheap to own in terms of total cost of ownership and easy to maintain, but this is clear only after you have been educated on many different aspects of the matter of car ownership. Although Toyotas are expensive to buy, they pay off in the long term, and have low total cost of ownership. It is not surprising then that Indians in America, given their high level of price sensitivity, like to own Toyotas.

The Oracle Asks:

Why are Toyota cars of such good quality? Why are shipping containers sometimes sent halfway across the world half full? Why do clothing stores such as Pantaloon and J. C. Penny have so many extra trousers sitting around on shelves? If the average expected sales of iPads is 100 units per month, does it make sense for a store to have more than a hundred tablets in stock? These and many other questions may be answered using operations management techniques.

Rubik's hypercubes

I came across the four and five dimensional versions of the Rubik's cube the other day. Absolutely fascinating. Links below.

• Four dimensions
• Five dimensions

Visual acoustics

Visual acoustics : "Paint" music with your mouse.

GapMinder

Here is a neat tool to visualize international development statistics. Notice the progress of the India blob versus time for various years.

Gapminder is a non-profit venture for development and provision of free software that visualise human development. This is done in collaboration with universities, UN organisations, public agencies and non-governmental organisations. [Link]

ClickDragType

My apologies in advance for bringing this to your attention. Another seriously cool Friday evening puzzle : ClickDragType. In the interest of not giving anything away, I will simply link to the site and leave it that.

Big Spanish Castle

Here is an optical illusion created using nothing beyond Adobe PhotoShop. Really neat.

Stare at the dot for 30 seconds. Then, without moving your eyes, move the mouse over the image. The image will look like it's in color until you move your eyes. [Link]

Friday game

Here is another Friday game. Check it out here.

The game involves four different particles falling from the top of the screen, which all look and move similar to sand. The particles resemble sand, water, salt, and oil in color. Along with these four, additional elements can be placed on the screen with the mouse, some that are solid and stationary instead of flowing. By mixing the different elements together, many colorful designs, complex structures, and systems can be created.

Chopper game

Here is a game for this Friday evening. M. sent this in and, at the time, he had a best of 1522. My best so far is 304. You can't blame a guy for trying :)

Update : I have left that record in the dust. 342!