Maths Puzzle: Geometry

An equilateral triangle is drawn with each side = 10. With the base of the triangle as one side, a square is drawn (with each side = 10) below the triangle. What is the radius of the circle on which the top tip of the triangle and the bottom two vertices of the square lie?

Height of triangle = sqrt (100-25)=sqrt(75)

Take the origin to be the bottom left point on square (1) = 0,0

Bottom right point on square (2) = 10,0

Top tip of triangle (3) = 5, 10+ sqrt(75)  

Assume 5,b is the center (meaning the height of the center is "b" from the bottom side of square).

Square of Distance from center to points on square (=radius squared) = 25+b²

Square of Distance from center to top point on triangle = ((10-b) + sqrt(75)) ²  

Now solve for b.

Puzzle - Killer's Love

http://www.albinoblacksheep.com/text/funeral

Speed At Work

A friend of mine (LEH) is a teacher. While at school, she is usually busy teaching or correcting papers. One day she was correcting papers. Generally, she hardly looks up to see what is happening around her. It was late afternoon and teachers started to leave for home one after another. LEH continued her corrections unaware of what was happening around her. 

Around 6PM she looked up to see that she was alone in the room and saw there were birds chirping outside. She smiled to herself and went back to her work thinking she will finish her work in a few minutes and then leave. 

She finished and then turned to look outside the window. Her jaw dropped, she started shaking. It was pitch dark outside and eerie. No noise.

There was no one within 100 ft of her. She desperately wanted to run down from the staff room and away from the school to civilization where there was light and where there would be people.

She knew the principal would be working late. But she was too embarrassed to call and request the principal on the phone to come up to the staff room and escort her down. LEH was too proud. No, she would manage on her own.

But she had to close the windows and lock the room before leaving. The school where she works is rather strict in these matters. She tiptoed to the window and in 10 minutes bolted the windows (her hands were shaking terribly by now, she was imagining faces outside the window) and walked fast back to the door. She closed the door and in the dark found the right key and with a snap (of her teeth which were chattering rather loudly) locked the door and swiftly flew down the stairs to light and safety.

After a couple of days she recounted this incident to me. As per her request I am letting others know of this incident. Correcting homework is quite harmful to health.

What Qualities Make A Good Teacher

http://jan.ucc.nau.edu/~slm/AdjCI/Teaching/Teacher.html

This article contains the opinions of two award winning teachers. What each one says is different. I relate to what Peter Beidler says but I don't understand much what Sally Phillips says.

Beidler says that a teacher who models his teaching style on the characteristics valued by students will be a successful teacher. Here I wonder. Can someone "model" his style? Is the style not largely, if not fully, innate to a teacher? Perhaps if he rephrased what he said as "a teacher, whose teaching style is in sync with what students value, will be a successful teacher" I would be more comfortable.

Birds Of Like Feathers Attract

I had earlier asked "Is there a rule that will determine when "Opposites attract" and when "Birds of the same feather flock together"?

A friend and I were trying to answer this question. There are two kinds of love that a person has. 

1. A love for something, an attribute (I call this a strong attribute; say, honesty), that a person themselves espouses. The person may not tolerate any person who does not have that strong attribute. In cases of such attributes, the person will flock  together with people who share the same attributes. There are other attributes, weak ones, which a person likes (rock music, traveling) which may be personal preferences and in such cases the person may still like other people who do not share these attributes.
2. A love for something, an attribute (say, tolerance, kindness) that a person needs and themselves don't have. Such a person will repel other people who are similar to themselves.

What does this tell us about ourselves? 

When we find that we cannot spend a lot of time with another person, either the other person:

• doesn't share an attribute that we have or 
• the other person doesn't complement us; they don't have that thing that we also lack.

This list of things that the other person, whom we dislike, lacks tells us about what we have (and need in the other person) and what we don't have (but expect the other person to have / provide it). The list has many attributes not all of which are equally important to us.

We do not give the same weight to all the attribute that we seek in another person. For example, in the movie Six days and Seven Nights, the man lived in a remote island and the woman was based in New York. The thought of being together was very difficult to digest initially. Subsequently they each figure that the other person had many other things which made their conflicting location preferences seem like a much smaller issue. (We do know what happened after they were together for 3 years. That is a very different story!)

Additional Reading:

• https://www.diigo.com/list/jazmynledford/annotated-bibliography-wp2
• http://www.examiner.com/article/do-opposites-attract-or-do-birds-of-feather-flock-together

Logical Puzzle

http://interviewkiller.com/question/1045-statements-all-rats-are-cows-no.html

Is there anything wrong with the options a, b, c, d?

Maths Puzzle

https://m.facebook.com/photo.php?fbid=635822939765412&id=489817594365948&set=a.490144640999910.130029.489817594365948&from_feed=1&refid=7&_ft_=qid.5870085079935915066%3Amf_story_key.-3908465476357452283

Ramanujam Maths Puzzle

I got this puzzle today: http://www.imsc.res.in/~rao/ramanujan/newnow/q289.htm

(sweat). Still working to find a formal answer to both the puzzles. Jesus. 

Elementary nested root problem, the author says. My foot.

Answer (inelegant) in green in this sheet: https://docs.google.com/open?id=0B1xQDlpMnLKxRUV6SExvdGRwWDQ

Formal solution here: http://www.imsc.res.in/~rao/ramanujan/newnow/sq289.htm and below.

Additional reading:

• http://paramanands.blogspot.in/2013/06/congruence-properties-of-partitions-part-1.html#.Vpp7X5p95dg
• http://www.had2know.com/academics/nested-radical-square-root-calculator.html
• http://en.wikipedia.org/wiki/Nested_radical

Pros And Cons Of Buying New Vs Old Flat

A friend of mine was planning to buy a new flat. He gave me a link which the builder had given him that stated the cost of the flat and amenities like 2 covered parking, 24 hours power backup and what not. The cost of the flat was Rs1.5 crore (approx $268,000).

It was a 1100 sq ft apartment which had a UDS (Undivided share) of 567 sq ft. The land cost in this area is Rs 4 crore per ground (per 2400 sq ft). I worked out that the UDS cost for 567 sq ft is about Rs 93 lacs (about $166,000) and the premium he was paying (excess of market price excluding tax / registration charge over UDS price) was about Rs 57 lacs ($100,000). I treat the UDS price as a book value. The ratio of market price 1.5 crore / UDS 93 lacs is like the price to book value (P/BV) ratio.

I felt it was not a very good deal to pay one third of the total price as premium, it should be far lesser. Could he not look for a flat which had a higher ratio of UDS price to market price. My friend's contention was that the if he only bought the UDS, it is like buying land and it will not fetch rent. Having a construction and roof over the land made it livable. And this construction essentially becomes the premium that one paid. My friend and I both agreed that the annual rent of a residential property across India was about 3.5% of the current market value. This figure of 3.5% is from my own data based on what I have seen in Chennai. I am sure it applies to other Indian cities as well. 

My friend pointed out that the rent is 3.5% of the total current value of the property, UDS + premium. Very true.

But the point is, while the value of UDS keeps increasing (say @ 10% CAGR), the premium depreciates. All your battery and woodwork and A/c are depreciating assets. At the end of 20 years they may lose all their worth. My data shows that the premium probably gets a negative worth after about 25 or 30 years. 

If this is so, the premium that you pay fetches you 3.5% annually on a declining value. Bank deposit gets you more than 7% where the capital is protected. What sense does it make to pay a high premium value? Is it not a better idea to buy a real estate with very low premium, for example very old flats? The entire property value would fetch 3.5% annual return PLUS there is little / no depreciation. Two flats, one old and one new are compared and the detailed workings are in the sheet below.

Buying a flat with low premium (example: a very old flat), fetches you 1 to 1.5% more CAGR.

This is another article which talks about the same subject.

Maths Puzzle - 23

Here is a puzzle. Go ahead and solve it. It's plain Maths.