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.

November 14, 2010

The first one is from Mathematics Olympiad Problems (1996 Vologda). The probability ones are inspired from enrich.maths.org

• There are 3 boxes, each consisting of two balls. One of the boxes contains exactly two white balls, one consists of exactly two red balls, and the third one consists of exactly one white and one red ball. But while labeling the boxes, their labels got mixed up so that none of the boxes has the right label. Is it possible to identify  which  box contains what by just drawing exactly one ball from each box? Can it be done by drawing just two balls or one ball?
• Since one of the kids in our team is a president of his class, he needs to come up with creative ways to do fund raising. What could be a better way then doing some kind of lottery!  Mr. President decided to setup the lottery as follows. In a bag, he places 4 identical hockey pucks in a bag, and numbers them from 1 to 4. Each kid (there are 24 of them who wants to make fortune!) walks up to him, hands him $1, and makes a guess from 1 to 4, and draws the puck. If the guess works, that kid gets $2, otherwise looses his money for the fundraiser. Do you think whether Mr. President will raise any funds? (BTW - kids can't gamble!)
• In the above setup, kids are asked to guess two numbers, instead of one. Then they draw two pucks and if the numbers on the puck matches with their guess, they win. First they draw a puck - record what is the number, places this puck back in the bag and then draw the second one (i.e., with replacement). For example if I guessed 12, then either I draw (1,2) or (2,1), I win. Do you think whether President will raise more money by this scheme. What happens when it is without replacement?
• What is the probability that two kids in the same family have a birthday on the same day?  There are approximately 30,000 families in City of Waterloo with two kids. How many of them will likely have kids having birthdays on the same  day.  There are approximately 14,000 families with three kids. What is the probability that all three of them will have their birthday on the same day.
• You are given a right angled triangle. You need to draw a line parallel to its base, so that the triangle is split in two parts of equal area. Where should one draw the line?
• There is a study done in Ontario schools, in terms of whats the average number of hours/per week the students in Grade 9 spend on their homework. Here is the data: 40 % don't bother, 10% spend 1-5 hours, 10% spend 6-10 hours, 20% spend  11-15 hours,  18% spend 16-20 hours and the rest spend more than 21 hours. Whats the average time a grade 9er spend on their homework?
• Consider the pattern:  The first term when n=0 is 2, when n=1 it is 8, when n=2 it is 14,  when n=3 it is 20, ... Once you figure out this pattern, you will realize that it is a linear pattern. You can graph it and see that it is a straight line.  For a pattern which grows linearly, show that it is sufficient to have two consecutive values to find the pattern rule.  Consider a pattern which follows a quadratic growth, for example the pattern when n=0 it is 1, when n=1 it is 2, when n=2 it is  5, when n=3 it is 10, ...17, .. 26,... Show that it is sufficient have three consecutive values to determine the pattern rule.  How does the graph of these patterns look like?

Sunday, November 7, 2010

• Consider three unit disks (i.e., of radius 1) touching each other. I am interested in finding the area of the triangular cavity between the three disks. Draw the triangle joining the center of the disks, and find the area of the part of the triangle, which is outside of the disks.
• Consider a cube, which is painted red, and has side length of 4cms. This cube is divided into unit cubes (i.e., each of side 1cm). How many small cubes will have (a) exactly three sides painted red (b) exactly two sides painted red (c) exactly one side painted red (d) no sides painted red.  Repeat the same problem with the starting cube of side length 1cm, 2cm, 3cm, .. Whats the pattern?
• Ad's friend decided to hangout on Friday evening in the Ottawa Expo. They planned carefully and decided that they needed to spend $96 for all the fun activities - and decided to contribute equally. But on the most exciting day, some of his friends (being very unreliable) didn't show up.! Now the remaining ones had to contribute $4 more than what they initially planned! How many of them were eventually at the expo?
• Mr Every T. Wrong is expert in getting everything wrong! He was asked to multiply a number by 4 and then add  eight - and he landed up dividing by 4  and then subtracting 8. He got 1 as his answer! What should be the correct answer.
• The overhead water tank on the Moodie drive supplies water to whole of Barrhaven.  It requires 4 hours to fill it completely and  6 hours to empty it. Usually the water tank is getting simultaneously filled as well as emptied. How long will it take to fill it? Suppose it takes x hours to fill and y hours to empty, what should be the relationship between x and y. This is a typical producer-consumer problem! Assuming that x is less than y, then how many hours it will take to fill the tank completely in terms of x and y?
• Ad is taller than Rd by 25%. Then, by what percentage Rd is shorter than Ad?
• The height of parliament hill in Ottawa is 92 mts and the height of the National Art Gallery  is 60 mts. They are approximately 200 mts apart. Assuming that their base is at the same level,  what is the distance between their tops? 
• You have a bag consisting of 10 red and 10 blue marbles. What is the smallest number of marbles you need to draw to be sure that you have exactly three of the same color? What happens when you have 10 red and 15 blue? What happens when you have three colors - say consisting of 10 red, 10 blue and 10 green - and you want to draw three of the same color?
• Whats the right length of laces that you need for your ice skates?  You need about 20  cms  to be left after tightening your skates to tie them up. There are 10 eyelets on each side,  3cms apart, and the distance between  two consecutive eyelets on the same side is about 1 cms.
  

Sunday, October 17, 2010

Some more problems based on the NTSE Exam for Grade 8.

1. In a triangle, one of the angle is same as the sum of two others, and those two angles are in the ratio of 4:5. What are three angles?
2. In a rectangle, one side is more than the other side by 4 cms. If both the sides are increased by 3 cms each, the total area increases by 81 square-cms. What are the measurement of the sides of the original rectangle.
3. A rectangular gallery has sides in the ratio of 5:2, and its total area is 100 square-mts. What are the side lengths?
4. You need to find a two digit number, where sum of its digits is 12, and when the digits are reversed, the difference between the two numbers is 18. 
5. A very keen hockey player signed a contract, where he will get $20/per game which he plays, and -$5/per practice which he lands up missing. There is one practice and one game each day, and at the end of the 30 day season, the player was rewarded $450. How games this player played, and how many practices he missed.
6. For her school project, Ms. Chrome, decided to make Kaleidoscope.  She decided to make the one where the tube needs to have length of 30 cms, and its diameter to be 7cm. Of course, you roll a sheet of paper, to form this cylinder. What should be the dimensions of the sheet of the paper (assuming no wastage). Understand the relationship between the surface area of a cylinder and area of a rectangle via this problem.
7. In our houses, we have a huge hot water tank in the basement.  What is the surface area of this tank? How long the sheet of the metal must be rolled to make this tank?  How do we find, how much volume it holds?
8. Check the reasoning (a)  a cow looks, at all the dogs in the barn, and sees that they have  a tail. Cow looks at herself, and sees that she has a tail. She concludes that she is  dog! (b)  All humans are mammals, and all mammals are vertebrates.  Are all humans vertebrates? (c) It rains on every Monday. It is raining today. Today is Monday. (d) It hasn't rain today at all, so it can't be Monday. 
9. Show that the product of any two consecutive even numbers is divisible by 4? Is it divisible by 8?
10. What about the product of any three consecutive even numbers? Is it divisible by 16?