Sunday, January 23, 2011

January 23, 2011

  • Take some square sheets of paper (e.g. post it notes), and a board pin. Arrange first 3 sheets, so that they cover the maximum possible area, and each of them is supported by the single board pin. Try this with 4 sheets. Now comes the interesting part:  what will happen when you have 5 sheets, 6 sheets, ... 100 sheets?  How should the arrangement of these papers should  look like - just supported by a single board pin?
  • Viviani's Theorem: Consider an equilateral triangle, and take a point anywhere in its interior. Calculate the distance from this point to each of the sides of the triangle, and look at the sum of these distances. Take another point, and do the same calculation. Wow - both the sums are the same! In fact this is same as the height of this triangle!  Will some similar statement be true for other regular shape figures - like a square or a hexagon.
  • Take a soccer ball, and tie the rope along its equator. Lets say a superhuman did the same way around earths equator. Now we want to make the rope bit relaxed - say we want to ensure that the distance between the ball (as well as Earth) and rope is at least 5 cms. Whats the additional length of the rope we need for the ball and for Earth?  (Earths radius is approximately 6350 Kms.)
  • We will do some Sangaku Geometric Art shapes - for example three circles tangent to the same line etc.

Sunday, January 2, 2011

Sunday, January 2 2011

Some of today's topics are from the The Math Book by Pickover.

Zeno's Paradox: To get out of my house, I need to exit through the door. That means I Ned to travel the distance to the door. I first travel half the distance, then half of the remaining half, and so on! This is like traveling 1/2+1/4+1/8+.... This will mean I can never come out of my house!

Cantor's arguments: Is the size of the set of even numbers same as that the size of odd numbers, what about is it same as the size of natural numbers, integers, ....

Suppose you have a cube of side length l, and you know that it's volume is l^3. you want to construct a cube whose volume is double of the original one. What will be the side length of new cube? Try with some concrete examples - for example with side length 2, 3,4,9.

You need to place 9 balls in the plane, so that they form 10 rows, where each row consists of three balls.

The problem from Bakhshali Manuscript [350]
There are 20 people in a group, consisting of men, women, children. They in all earn 20 coins, where each men earn 3 coins, each women earn 1.5 coins and each child earns 0.5 coins. How many of each of men, women, and children are there in this group.

How many kilos of grains of rice on a chess board? Each grain of rice weighs approximately .020 grams. Here is the ancient puzzle, the first square consists of one grain, the next two, the next four and so on. The chess has 64 squares. The number of grains is approximate. A 20 digit number. What size of the bag we will need, when about 100 grains can be packed in a one cubic centimeter.