Saturday February 26, 2011

• Look at the figure on the left. Came from ancient Chinese Math (Zhou bi, 1045 BC onwards) )! It is drawn on a 7x7 square. Each triangle (yellow or green) is of dimension 3x4. The black square is of dimension 1x1.  Each triangle is right angled. Whats the area of the square made up of green triangles and black square. What's the side of this square - do you see the Pythogorean theorem! 
• You can try to do the same with outer square being of size 14x14, each triangle of dimension 6x8, and the black square of dimension 2x2.
• In general, you can prove the Pythogorean theorem as follows: assume that the sides of the right-angled triangle are a and b, and we need to show the hypotenuse is sqrt(a^2+b^2). Assume a is greater than or equal to b. Draw the above picture by taking the sides of the outer square to be a+b. The dimension of the inner square are a-b times a-b.   Now it should be straightforward to see that the area of the green square (inclusive of the black one) is  4*area of green triangles + area of the black square = 4*1/2*ab+(a-b)(a-b)= a^2+b^2, and hence the side of this square will be sqrt(a^2+b^2).
• This is not really a math problem - sort of related to do with string manipulation -You need to change WIDE to RISE, where the rules of the game is to change only one character at a time and each intermediate word is meaningful. Whats the smallest number of transformations you need to do? Try doing this from LOVE to RIFT.
• An outdoor swimming pool is 25ft by 50ft and is 8 ft deep. In the morning it is full of water, and by the end of the hot summer day, water drops down by 1.5 feet, due to evaporation. How much water is lost? How many buckets it is? What is the rate of evaporation - lets say we have 16 hours of sunlight in Ottawa in summer - but the peak is from 11AM till  7PM. How can we minimize the evaporation?
• The ratio of the number of goals between  Alfie and Alex is 3:4 and between Alex and Sid is 5:6.  Whats the ratio of goals between Alfie and Sid.
• Anant in his grade 5/6  class found the following stat when he conducted the chocolate poll.  In all 80% liked the chocolate. The ratio of Grade 5 to Grade 6 kids in his class is 2:3. What are the chances that when you pick a `random' kid from Anant's class - that this one really likes chocolate and is in grade 5?
• Four identical cubes are placed next to each other to make a rectangular prism. The surface area of this prism is 360 sq cms less than the sum total of the surface area of the  four cubes.  Can you determine the dimension of the cube?
• Next year my age and Mr. A's age  will be prime numbers, and the product of our ages will be 611. How old are we now? Of course, there is exactly one way to non-trivially factor 611, since its a product of primes. How will we do it, if in place of 611, its a very large number - for example a number made up of 500 digits! The computationally difficulty of finding factors of such large numbers lies at the heart of most of the secure transactions over the internet!
• Whats the last digit in the product of five consecutive numbers, where one of those numbers has 7 as its last digit.
• What is the smallest possible number that can be multiplied to 120, so that the product is a cubic number?