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?

 

Sunday, Febrauary 13, 2011

Problems are inspired by the Grade 9 text book today!

• Typically, in bikes (cycles), there is a plastic reflecting light which is attached to a wheel, so that the bike is visible to a car coming across. What  will be the shape of the path traversed by this light, when the bike moves along?
• Mr. Forget started  to walk back to his home from the grocery store, located 650 mts from his home. He started to walk back in the direction of his house, but went 800 mts, then suddenly remembers that he has gone past, and switches direction, and then walks back half the distance this time (400 mts), OOPS, he again remembers and then switches direction and walks back, and this continues .. Will he ever reach his home?
• What is the total distance a puck travels in a hockey game? How will go about estimating this distance.  Ice rink is 200ft by 85 ft?  Assume that the hockey game is 60 minutes long, an average pass is approximately 15ft.
• You  take a very long string, and double it up, then redouble it, then reredouble it, and so on. Lets say you did it 50 times. Now cut the string from the middle. How many pieces will you get?
• How will you show that the sum of three consecutive numbers is divisible by 3. Can one make the same statement for 4 consecutive numbers being divisble by 4 and so on. It of course doesn't work for two.
• Lets do some fun experiments with the Mobius strip. Take a ribbon, twist and then tape it. Mark the center line - and cut along, see what happens?  What if you have two lines - say 1/3rd and 2/3rd away and then cut along them, and see what happens?
• Half of a fraction, decreased by 0.75 results in 7/12. What was the fraction?
• How many different bracelets can be made consisting of 2 red and 3 green beads?
• What is the smallest and largest possible value of a/b + c/d, where a,b,c,d need to take values from {1,2,3,4}?
• What will be the last digit in 2^1234?

February 6, 2011

• Dr. A. for the sleepover party teased his friends. While they were sleeping, he put a pink sticker on the forehead of all of his friends. When they woke up, they all started laughing looking at each other! Then suddenly Mr. B stopped laughing - Why?
• How can we find the surface area of a tetrahedron whose sides are of length 1. What about an octahedron? What about their volumes?
• Mr A. has two types of cards in his pocket. One card is red on both the sides and other one has red on one side and green on the other side. He takes a card out - and sees that one side is red - what are the chances that the other side is red as well? (50% is not the right answer BTW)
• How heavy is the water mattress? Recall that Queen mattress dimensions are 152x200x20 cms and 1 cubic meter of water weighs 1000Kilos.
• How to measure volume of a stone? Suppose you have a beaker (which is a cylinder). Lets say you fill the beaker of radius 6cms with water, and the height of the water in the beaker reads 14.2 cms.  Now gently drop the stone in the beaker and water rises to 18.7 cms. What is the volume of the stone? 
• Next set of questions are from Grade 9 EQAO testing

1. If x=1/3, then what is 6x^2.
2. Typically car sales (and many other sales)  are paid a fixed amount per week and certain percentage of their sales. Lets say that this person earns $500/week and 2.5% of the sales. If  the total payment for the week was $700, then how much were the sales in that week?
3. What is the sum of the interior angles of a regular pentagon; hexagon; 12 sided figures?
4. You ordered CDs from your favourite store online. Each CD costs $11.44 + Tax. Total you paid is $90.49 which includes HST.  How many CDs did you buy?

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

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.

Sunday, Boxing Day, 2010

Todays problems are based on SAT Exams.

1.  On the boxing day, the sale price of all the items in the Eagle was dropped by 20%.  By what percentage they need to be raised, to bring them back to the level they were before the boxing day.
2. If a number x is divided by 7, the remainder is 5. When the number y is divided by 7, the remainder is 3. What is the remainder when x+y is divided by 7.
3. The average of a set of 5 numbers is 30. The average of three of them is 24. What is the average of the other two numbers?
4. Given two numbers x and y, find the average of (x+y)^2 and (x-y)^2. Check your answer by setting x and y as (3,4) , (3-3), ...
5. Mr. Generous wants to buy the best possible home theater for his family, so that  they can enjoy the vacation by watching best of the best. He goes to future shop, and asks the Salesperson to help him. He needs to buy a TV, DVD Player, Speakers and  Amplifier. In all Future Shop carries 20 types of TVs, 10 types of DVD players, 16 types of speakers and 6 types of Amplifiers. Each option takes him approximately a minute to evaluate. How long you think it will take him to make the best possible judgment?
6. In Ottawa, all telephone numbers are listed as 613 xxx xxxx. How many different telephone numbers are possible? What about Toronto - do you think that one area code will suffice?
7. Ad was assigned the job of lining up 5 kids, ages 2,3,4,5, and 6 in a Q, according to their age, but they are restless - and they keep shuffling their order in the line. What is the chance that Ad will succeed?   What is the chance that at least 4 of them are in the right order? What about at least 3?
8. If my BBry buzzes every 5 mts and my iphone buzzes every 7 minutes, then when is the first time they will buzz simultaneously, assuming they were turned on at the same time? What if it was 4 and 8 minutes? What this had to do with prime numbers?
9. My car typically travels about 500 Kms/week, and out of that 1/5th is on Highway and 4/5th on the city roads. For a liter of a gas, it gives an average of 9Kms in city and 11Kms on Highway. The cost of Gas today is $1.15/liter. How much money should I spend on the gas in a week?  The new Honda Hybrid  has the rating of  20Kms/l in city and 22.5Kms/l on Hwy. Whats the cost per week for the Hybrid?
10. This is based on  the TED talk which I heard recently - are more choices good or bad for us? Think about the example of the Home Stero, suppose we had only two choices for each in place of so may possibilities, then what would you have preferred. Think of you set it up - and then you didn't like it - who will you blame?

Sunday November 21, 2010

• A thick metal pipe,  2mts long, has inner radius of 12 cms and outer radius of 15cms, is made of steel. How much volume of steel is used to make this pipe?  What do you think is the weight of this pipe? Note that weight of 1 cubic meter of steel is 8000 Kg.  What if the pipe is made of Bamboo - 1 cubic meter of Bamboo is approximately 350 Kg.
• The new Chapman Mills park is rectangular, where the sides are in the ratio of 4:3. The total area of this park is approximately 3500 sq mts. What is the cost of installing a metal fence around this park, where The Ottawa Fencers charges approximately $45 per linear meter.
• Ms. Quick finishes a job in 8 days, whereas Mr. Slow finishes the same job in 12 days. How long will it take if both of them will work together (assuming that the job can be partitioned nicely (like pulling weeds in the garden)).
• Dr. Light needed to climb his wall to install the newly bought Christmas lights. His ladder is 8mts long, and the wall is approximately  7mts long. How far is the base of the ladder from the wall? Do you think its at a safe distance and will he be stable on the ladder?
• Suppose we toss two unbiased coins 100 times and observe that the number of times we get 2 heads is 25 and number of times we get no heads is 18. How many times we got at least 1 tails? We can try the same problem with 3 coins - lets say #times 3 heads= 22, #times 2 heads=32, #times 1 heads = 24, #times 0 heads = 22.   How many times we get at least one tails, at least one tails and one head.
• A disc needs to be cut out from a square sheet. The length of the sheet is 2mts. Whats the area of the largest possible disc? Whats the area of the left out piece of the sheet? Try the same problem for a cube and a sphere. Is the volume of the left out piece larger than that of the sphere?
• The angles of a triangle are in the ratio of 1:2:3. What are the angles?  Is it a right angle triangle? What if it is 3:4:5?
• Mr. Too Quick puts all his favorite NHL-team socks in the drawer, and in the morning rush (in the dark in long Northern Winter)  - what are chances that he will draw a pair which is of the same team. He has 2 pairs of SENS, 2 pairs of PENS, 2 pairs of  DUCKS and 3 pairs of HAWKS. What are the chances that he will draw a pair which does not belong to the same team?  First try this for two teams, and lets say one pair from each team, and then start to make it more complex.