November 20, 2011

Today's problems are inspired from the set of Problems from the Waterloo's Centre for Education "Problem of the Week" collections.

1. Consider the above figure. Assuming that the angle BAC is x-degrees, and |AB|=|BC|=|CD|=|DE|=....   How many triangles like ABC,  BCD,  CDE, DEF, .. one can form?  Of course this will depend upon the angle x. Try it for x=15, 30, 45, .. (Smaller angles are more interesting than the large ones!) 
2. How many different numbers are there between 10 and 99, so that the sum of their digits equals 10? Try for numbers between 100 and 999?
3. Take a right angled triangle, say with vertices ABC, where |AB|=3, |AC|=4, |BC|=5. Imagine that this triangle is standing on the edge AB, and you rotate it full circle (360) by keeping A fixed, but moving B the full circle. What is the volume of the swept figure? What if the triangle is standing on just the vertex A, and you rotate it full circle. What will be the volume of the swept figure?
4. There are six houses on a street, where the average income of 1st and 2nd is 70k, 2nd and  3rd is 80k, 3rd and 4th is 90k, 4th and 5th is 100k, 5th and  6th is 80k. What is the average income of this street? Whats the average income of 1st and 6th? If I tell you that the first household made 80k, then can you determine the income of each of the house? [In many survey statistics, they  collect several types of information - family income, age, ethinic group, education qualification, etc., and then one can query the database in terms of finding out average statistics about a neighborhood.  For example you can look at study of immigrants in Ottawa Area.  There is a possibility that by looking at different statistical tables, you may be able to gather some extra information. For example, from the study of immigrants, a grocery store in a particular locality may carry more Caribbean delicacies, or Chinese, or South Asian. Similarly knowing the average age and income, the marketing and advertisement may be targeted. In our problem, if I didn't tell you that the first house made 80k, what extra piece of information can help me to deduce the income of each of the families?] 
5. Consider a five digit number abcde. The digits 1, 2, 3, 4 and 5 are used to form this number (all digits are being used). You need to find what is the number, and the extra information  provided to you is that abc is divisible by 4, bcd is divisible by 5, and cde is divisible by 3. [Recall that a number is divisble by 5, if the last digit is either a 0 or 5. A number is divisible by 4 if the last two digits are divisible by 4, and a number is divisible by 3 if the sum of its digits are divisble by 3.]
6. A number is split into several equal parts in the following way. 1st part is 10 + 10% of the remaining. 2nd part is 20 + 10% of the remaining now. 3rd part is 30 + 10% of the remaining now. 4th part is 40 + 10% of the remaining now. and so on.  If each  part has the same value, then how many parts are there, and what is the value of each part? You can try this with 20% in place of 10% and see what happens?
7. Mr A., being an A student, received 94% marks average among the six subjects in his mid-term exam. But just before he was supposed to receive the great honor from the School's principal, his French teacher realized that his marks in place of being 96/100 should have been 69/100. (May be the teacher read them upside down!). Whats Mr A's right average then?  
8.  Two airplanes are trying to land at Ottawa Airport on the same runway! The control tower needs to determine how to ensure that they can safely land. Control tower knows which aircrafts are coming, and it knows the location of the two aircrafts in the sky (say their distance and what angle they are with respect to some reference). The tower needs to tell the aircrafts what path to follow,  ensuring that they are separated by 5 Nautical Miles (approximately 10Kms) from each other at any point of time till they land. Can you think of some strategy for figuring out these paths. Now think about worlds busiest airports like the one in Atlanta or in London or in Chicago.  Approximately 200 flights/hour arrive in Atlanta (thats 3 per minue!).

Sunday, June 26, 2011

• Given a unit square, what is the ratio of the areas of two circles - the circle with maximum area lying completely inside the circle, and the area of the smallest area circle that completely encloses the square. 
• How to do the same problem with a cube and inscribing and circumscribing  sphere?
• Given a unit square, you want to place a triangle whose area is maximum that lies completely inside the square. What will be its area?
• How to do the same problem with a triangle in the cube?
• How many corns on a corn? Typically a corn is cube shaped  measuring 4mm, and a corn cone is of diameter of 7cms and has a length of 27cms.
• Show that in a circle, if we take two chords of the same length, then the angle they form at the center of the circle is the same. 
• Show that in a circle, the line joining the center of the circle to the mid point of any chord, is perpendicular to the chord.
• How many circles pass through a single point? Two points? What about three points? Show that if three points do not lie on a straight line, there is a unique circle passing through them.
• Show that the angle subtended by any arc of the circle at the center is double of the angle subtended by this arc at any point on the circle.

Sunday, May 8, 2011

Some of these problems are based on triangle inequality.

Draw triangle whose sides are of length 4,5 and 7 cms. Try doing the same exercise with sides of length 3,4, 8 cms. Why can't you make a triangle in the 2nd case?

Show that in any triangle, the longest side is always smaller than the sum of other two sides.

Given three positive numbers a, b, c in a non-decreasing order, such that c is at most a+b, then you can always form a triangle with side length a, b and c.

Show tha no side of a triangle has a length larger than half of it's perimeter.

Show that in a convex quadrilateral (no interior angle bigger than 180 degrees), the sum of the length of two opposite sides is no larger than the sum of the lengths of diagonals.

Where to place the Strandherd-Armstrong bridge?
Suppose that Rideau River is a straight line, and two friends live on opposite side of the river. Where should the new bridge be placed so that their travel time is as small as possible. What if, there are two friends on one side, and one on another, what if two and two, and what if two communities?
what if the city is rich enough to approve two bridges?

The distance between big apple on Hwy 401 to Montreal is 400 kms, and to Toronto is 160 Kms.
Does that mean that the distance between Montreal and Toronto is 260 kms? Also, the distance to Ottawa is about 350kms. Does that mean that Montreal to Ottawa is 50kms. In general, if you are given distances from a point to some of the cities, what can you tell about minimum and maximum distances between these cities.


What will happen if we have the following variation - Sum of the two sides is always smaller than the third, i.e, To go from a to b, it is always best to go via c! In other words, it's always best to take a detour. Can we ever reach b from a?

Saturday April 16, 2011

This is federal election time in Canada as well as budget talk in the US. Questions are based on those!

• The total US debt  as of March 25, 2011 is  14.26 trillion dollars. How much is 1 trillion? (Just to put this number in perspective - total public parking spaces (metered/road) etc. in San Francisco is about 500,000. Average cost of a car in that area is about $30,000. Compare the debt in terms of some quantity you can imagine.
• Here is an alternate way to understand this figure. The population of US is approx. 311,000,000. Whats the debt per person then? per family? What will a family do if they are in such a debt?
• Canada's debt is approximately 540 billion dollars, and population is 33,739,000. Compare the debt per person and per family v/s US. Growing economy like India has a debt of approximately 750 billion $s. Population is 1.2 billion. 
• What is debt? This is essentially what govt. owns to people (shares/bonds...). Usually it is seen in comparison to GDP, gross domestic production. This is the market value of all the goods and services produced within a country in one year.  Canada's GDP is about 1.6 trillion $, debt around 540 billion $. US GDP is 14.6 trillion and debt 14 trillion. (How can Prime Minister/President reduce debt and increase GDP - whats a healthy economy?) 
• During the federal election in Canada, it is claimed that the cost of election is $300 million dollars. Calculate whats the cost of election per household in Canada? We are having almost one election/year - as opposed to one in 5 years.
• In 2008, in Canadian elections, there were approximately 23,000,000 voters on list and only 14,000,000 voted. The distribution of votes in this election between main parties (Conservatives, Liberal, Bloc, NDP, Green) were  (38%,27%,10%,18%,7%) and the number of seats in the parliament were (143, 77, 49, 37, 0), respectively.  What do these numbers mean? Does parliament reflect the % of votes? Can you think of a better mechanism?
• If you hear the results on May 2, after the polls close, you will see many types of graphs, analysis, trends, ups and downs etc.   Can you think of drawing a line graph whose slopes are 1, 0, -1, 2, 3, -2, -3.  What linear equations these graphs satisfy assuming that they pass through the origin.  
• If I have a line segment, lets say originating at (0,2) and ending at (4,13), then I can measure its length by actually drawing it and then use a ruler. Can I do in some other way as well?  In general if the coordinate of the endpoints are (x1,y1) and (x2,y2), what is its length?

Sunday April 10, 2011

1. On the Boxing Day (26th Dec.), stores need to come up with curious ways to give discounts to attract customer. Here is an interesting one - the store is offering a discount of  an additional 10% every hour  -  what will be the price of an item worth $100, at 9AM, 10AM, 11AM, ..., 5PM. Think of - it the discount is on the original price or the last listed price. Since you know that there are limited quantities of each of the items,  how should you approach?  If all the customers can cooperate what could be your strategy - what if they do not want to cooperate at all?
2. In 10 seconds, the distance that a cheetah, cyclist, and an Olympian  runner can cover are respectively 300m, 160m and 110m. What is their speed? How much they would have covered in 8 seconds or 12 seconds - assuming a constant speed? Whats the best way to find out the distance for any given time duration?
3. A right angled triangle with base 4mts, aligned to x-axis, has an area of 6m^2. What is the slope of its hypotenuse? 
4. Whats is the slope of a Standard Staircase?  (By the way the code for staircases is 7-11, i.e. 7 inch rise for each 11inch.)  Typically the 1st floor ceiling is about 9ft,  how many steps will it require?  Whats the linear distance one needs to reach 9ft, if each step is 11inch deep. Many places steps are made in L, U or semicircular shape - any idea why one builds these shapes?
5.  Last few questions dealt with the slope of a straight line - which is defined to be `rise' over `run'. What can one say about slope of a curve? or a surface? How will one go about computing slope of a curve?
6. Each electron has a charge of  1.6 * 10^-19 Coulomb.  How many electrons you need to make a charge of 1 Coulomb?
7. Current in electrical circuits means how much charge passes through a conductor in 1 second. For example, current of 1 Ampere means a charge of 1 Coulomb has gone through the conductor in 1 second - think of how many electrons have moved through - traffic jams?   If the fuse in the house has a rating of 10 A, then how much charge has gone through the fuse in 1 second? How many electrons? 
8. My house has an electric panel and it is rated as 100 Amp. What does that mean? Lets take any household appliance - for example a computer - and lets look at its ratings? Can we figure out how many computers can we run on a 100 Amp circuit, where we have 110 Volts. What about some more heavy duty appliance like - drier or stove. For example stove are rated for 20Amps.

Sunday April 3, 2011

• 

•  The above graph  (taken from a Forex Blog) shows the trend of CAD/USD loonie over last  5 years. Any conclusions can you draw?
• This is year of India in Canada. Indian Government decided to take up a survey to see how are Indian's performing outside India. They needed to compare Indians living in `similar' type of countries - for example US, Canada, UK and Australia. What should be a good hypothesis to test? What kind of primary/secondary data can be used? What kind of conclusions can be drawn? Is there any point of doing this kind of study?
• Mr. Ad. needs to form a team of 30 players who will participate in several of summer sports (sort of mini summer olympics) where the sports include running, field hockey, tennis, soccer, jumping, swimming, etc. What selection criteria should he use (name at least 5).  Design some hypothesis to test whether the selection criteria leads to medals. What kind of data primary or secondary can be used to verify his hypothesis.
• Here is a chart showing  (taken from here) Speed v/s Safe stopping distance in icy v/s normal conditions. Why do you think its not a straight line curve?

• This web-site lists some of the super cars, their price and the time (in seconds) it takes them to reach the speed of 100Km/hr.  For example, 2005 Ferrari FXX reaches that speed in 2.5 secs, and it costs $1.5 million.  What kind of plot do you expect in terms of price v/s time it takes to attain 100Km/hr. (For example my car takes 7.4 secs - though I never tried that).
• What kind of distance-time graph you will expect when you hit a Home run in Baseball - the launch of a satellite - a train entering a station to stop - tiger chasing its kill - in general a typical commute from home to office  (In distance time graph, you will plot the distance of the ball (or an object) from its original position as time increases).
• How many odd 1-digit, 2-digit, 3-digit, 7-digit numbers can be formed using the digits 1,2,3,4,5,6,7, if each number consists of distinct digits (e.g. 223 is not valid!).
• This one is based on Zero Knowledge Proofs - its an interesting  concept.  See wikipedia entry on this (this picture is from there) Idea is pretty simple. Both of these persons don't trust each other. Person standing outside the cave (call him Bob), needs to know the number-key of  the door which is at the far end of the cave. Person standing in the cave (call her Alice) claims that she knows the number key, and is willing to give that key for $100. Bob, can gamble, and pay Alice $100, and hope that she is telling the truth. But she may not!  How can Alice convince Bob that she knows the key, without revealing the key, especially before Bob pays her $100! This kind of technique is used, for example the password you have on your bank card.

 

Sunday, March 27 2011

• Why does long division works? This is an exercise in number representation. Divide 123456 by 11 - explain why your method works? Divide 10000250 by 10 and then explain why your method works? Did your reasoning hold with consecutive zeros?
• Now try dividing x^3+x^2-3x+1 by x-1. Follow the same steps, and try to get rid of the highest powers of x in each step. Cross check your solution by multiplying your result with (x-1).  Did the same reasoning hold for long division?
• Mr. A has 30% sens card, 25% pens card, 15% ducks card, and rest of them are equally distributed among 27 different teams. He has been buying 4 cards per week, costing him 50c/week. He does this whenever he plays his hockey game. The season in all consists of 32 games - how much money he spent - how many sens, pens, and ducks card he has?  How many HABS card he has? What are chances that he has a PK Subban card? How much money he should invest to be more or less certain that he has a Subban's card?
• Consider a square whose each side is of length one (unit square). How long is its diagonal? How will we do this for a unit cube?
• Lets do the above problem for a cylinder, whose base has a radius of 1.5m, and its height is 4m. How long is the diagonal.
  

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.