tag:blogger.com,1999:blog-60236214750952465392018-03-02T11:43:03.258-05:00Math Problems for Middle School StudentsI have been running a "Math Class" for a few kids, in Grades 5/6/7/8 for an year now. We usually meet once a week for an hour. The idea is to do problems which are not part of the curriculum - problems which require thinking and reasoning. These problems are picked up mainly through internet. Once I get used to blogging, I will try to provide appropriate references. Any creative comments, suggestions and references will be very helpful (email to anil@scs.carleton.ca).Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.comBlogger27125tag:blogger.com,1999:blog-6023621475095246539.post-55107720587947269012011-11-20T18:20:00.000-05:002011-11-20T18:20:15.908-05:00November 20, 2011<div dir="ltr" style="text-align: left;" trbidi="on">Today's problems are inspired from the set of Problems from the Waterloo's Centre for Education "Problem of the Week" collections.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-sxdoPJV6CdE/Tsl1rnWUM-I/AAAAAAAAABk/XhVTJmkc7aA/s1600/triangle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="134" src="http://3.bp.blogspot.com/-sxdoPJV6CdE/Tsl1rnWUM-I/AAAAAAAAABk/XhVTJmkc7aA/s320/triangle.png" width="320" /></a></div><ol style="text-align: left;"><li>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!) </li><li>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?</li><li>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?</li><li>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 <a href="http://www.ottawa.ca/city_services/statistics/counts/counts_jul_04/index_en.shtml">study of immigrants in Ottawa Area</a>. 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?] </li><li>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.]</li><li>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?</li><li>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? </li><li> 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!).</li></ol></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com35tag:blogger.com,1999:blog-6023621475095246539.post-5512936461323030742011-06-26T16:12:00.001-04:002011-06-26T16:12:55.222-04:00Sunday, June 26, 2011<div dir="ltr" style="text-align: left;" trbidi="on"><ul><li>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. </li><li>How to do the same problem with a cube and inscribing and circumscribing sphere?</li><li>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?</li><li>How to do the same problem with a triangle in the cube?</li><li>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. </li><li>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. </li><li>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.</li><li>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.</li><li>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.</li></ul><br /></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com1tag:blogger.com,1999:blog-6023621475095246539.post-67249068123566228012011-05-08T12:42:00.000-04:002011-05-08T16:26:06.233-04:00Sunday, May 8, 2011Some of these problems are based on triangle inequality.<br /><br />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?<br /><br />Show that in any triangle, the longest side is always smaller than the sum of other two sides.<br /><br />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.<br /><br />Show tha no side of a triangle has a length larger than half of it's perimeter.<br /><br />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.<br /><br />Where to place the Strandherd-Armstrong bridge? <br />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? <br />what if the city is rich enough to approve two bridges?<br /><br />The distance between big apple on Hwy 401 to Montreal is 400 kms, and to Toronto is 160 Kms.<br />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.<br /><br /><br />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?Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com1tag:blogger.com,1999:blog-6023621475095246539.post-77191785258892467972011-04-16T17:16:00.002-04:002011-04-16T17:20:37.407-04:00Saturday April 16, 2011<div dir="ltr" style="text-align: left;" trbidi="on">This is federal election time in Canada as well as budget talk in the US. Questions are based on those!<br /><br /><ul style="text-align: left;"><li>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.</li><li>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?</li><li>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. </li><li>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?) </li><li> 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.</li><li>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?</li><li>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. </li><li>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?</li></ul><br /><br /><br /><br /><br /></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-38458014692213192442011-04-10T16:59:00.000-04:002011-04-10T16:59:35.404-04:00Sunday April 10, 2011<div dir="ltr" style="text-align: left;" trbidi="on"><ol style="text-align: left;"><li>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?</li><li>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?</li><li>A right angled triangle with base 4mts, aligned to x-axis, has an area of 6m^2. What is the slope of its hypotenuse? </li><li>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?</li><li> 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?</li><li>Each electron has a charge of 1.6 * 10^-19 Coulomb. How many electrons you need to make a charge of 1 Coulomb?</li><li>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? </li><li> 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.</li></ol></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com6tag:blogger.com,1999:blog-6023621475095246539.post-77970561640354816262011-04-03T11:59:00.000-04:002011-04-03T11:59:46.490-04:00Sunday April 3, 2011<div dir="ltr" style="text-align: left;" trbidi="on"><br /><ul style="text-align: left;"><li><a href="http://4.bp.blogspot.com/-_CHjj_kD_SI/TZh-PpiDliI/AAAAAAAAABU/AMchh-QciQs/s1600/CAD-USD-5-year-chart.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="225" src="http://4.bp.blogspot.com/-_CHjj_kD_SI/TZh-PpiDliI/AAAAAAAAABU/AMchh-QciQs/s400/CAD-USD-5-year-chart.png" width="400" /></a></li></ul><ul style="text-align: left;"><a href="http://3.bp.blogspot.com/-JYQ3ZnumP3Q/TZiDGHqzt_I/AAAAAAAAABY/Pg_NTmGDjcI/s1600/150px-Zkip_alibaba1.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><li> The above graph (<a href="http://www.forexblog.org/category/canadian-dollar">taken from a Forex Blog</a>) shows the trend of CAD/USD loonie over last 5 years. Any conclusions can you draw?</li><li>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?</li><li>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.</li><li>Here is a chart showing (<a href="http://www.drivingfast.net/techniques/winter-driving-techniques.htm">taken from here</a>) Speed v/s Safe stopping distance in icy v/s normal conditions. Why do you think its not a straight line curve?</li></ul><a href="http://2.bp.blogspot.com/-k2cQBTwKY4Y/TZiIBVWDVCI/AAAAAAAAABc/rjzTiFgjqcU/s1600/winter-stopping-distances.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="376" src="http://2.bp.blogspot.com/-k2cQBTwKY4Y/TZiIBVWDVCI/AAAAAAAAABc/rjzTiFgjqcU/s640/winter-stopping-distances.gif" width="640" /></a><ul style="text-align: left;"><li>This <a href="http://sites.google.com/site/investincars/exotic-supercars/super-car-0-100km-h-times-verified">web-site </a>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).</li><li>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). </li><li>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!).</li><li> This one is based on Zero Knowledge Proofs - its an interesting concept. <a href="http://en.wikipedia.org/wiki/Zero-knowledge_proof">See wikipedia entry on this (this picture is from there)</a> 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. </li></ul><a href="http://3.bp.blogspot.com/-JYQ3ZnumP3Q/TZiDGHqzt_I/AAAAAAAAABY/Pg_NTmGDjcI/s1600/150px-Zkip_alibaba1.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"> <img border="0" height="280" src="http://3.bp.blogspot.com/-JYQ3ZnumP3Q/TZiDGHqzt_I/AAAAAAAAABY/Pg_NTmGDjcI/s400/150px-Zkip_alibaba1.png" width="400" /></a></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-72967710542587540682011-03-27T10:04:00.006-04:002011-03-27T16:53:26.178-04:00Sunday, March 27 2011<div dir="ltr" style="text-align: left;" trbidi="on"><ul style="text-align: left;"><li>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?</li><li>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?</li><li>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?</li><li>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?</li><li>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.<br /></li></ul></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-48591217402651213602011-02-27T12:43:00.005-05:002011-02-27T14:32:44.828-05:00Saturday February 26, 2011<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="https://lh6.googleusercontent.com/-eQ6LPZniz0g/TWlHeMBMu8I/AAAAAAAAABQ/BYCVynpkSP4/s1600/chinese.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://lh6.googleusercontent.com/-eQ6LPZniz0g/TWlHeMBMu8I/AAAAAAAAABQ/BYCVynpkSP4/s200/chinese.png" width="200" /></a></div><ul style="text-align: left;"><li>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! </li><li>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.</li><li>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). </li><li>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.</li><li>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?</li><li>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.</li><li> 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?</li><li>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?</li><li>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!</li><li>Whats the last digit in the product of five consecutive numbers, where one of those numbers has 7 as its last digit. </li><li>What is the smallest possible number that can be multiplied to 120, so that the product is a cubic number? </li><li> </li></ul><ul style="text-align: left;"> </ul><ul style="text-align: left;"></ul></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-90038153370197689792011-02-13T15:34:00.000-05:002011-02-13T15:34:35.014-05:00Sunday, Febrauary 13, 2011<div dir="ltr" style="text-align: left;" trbidi="on">Problems are inspired by the Grade 9 text book today! <br /><ul style="text-align: left;"><li>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?</li><li>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?</li><li>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. </li><li>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?</li><li>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.</li><li>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?</li><li>Half of a fraction, decreased by 0.75 results in 7/12. What was the fraction?</li><li>How many different bracelets can be made consisting of 2 red and 3 green beads?</li><li>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}?</li><li>What will be the last digit in 2^1234?</li></ul></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-8898684460525622182011-02-06T16:14:00.000-05:002011-02-06T16:14:30.418-05:00February 6, 2011<div dir="ltr" style="text-align: left;" trbidi="on"><ul style="text-align: left;"><li>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?</li><li> How can we find the surface area of a tetrahedron whose sides are of length 1. What about an octahedron? What about their volumes?</li><li>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)</li><li>How heavy is the water mattress? Recall that Queen mattress dimensions are 152x200x20 cms and 1 cubic meter of water weighs 1000Kilos.</li><li>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? </li><li>Next set of questions are from Grade 9 EQAO testing</li></ul><ol style="text-align: left;"><li>If x=1/3, then what is 6x^2.</li><li>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?</li><li> What is the sum of the interior angles of a regular pentagon; hexagon; 12 sided figures?</li><li>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?</li></ol></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-65831166967501842972011-01-23T18:00:00.000-05:002011-01-23T18:00:28.547-05:00January 23, 2011<div dir="ltr" style="text-align: left;" trbidi="on"><ul style="text-align: left;"><li>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? </li><li>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. </li><li>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.)</li><li> We will do some Sangaku Geometric Art shapes - for example three circles tangent to the same line etc. </li></ul></div>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-42227058045286428922011-01-02T17:49:00.000-05:002011-01-02T17:49:01.868-05:00Sunday, January 2 2011Some of today's topics are from the The Math Book by Pickover.<br /><br />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!<br /><br />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, ....<br /><br />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.<br /><br />You need to place 9 balls in the plane, so that they form 10 rows, where each row consists of three balls. <br /><br />The problem from Bakhshali Manuscript [350]<br />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.<br /><br />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.Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-23757090544765144442010-12-26T16:37:00.000-05:002010-12-26T16:37:24.487-05:00Sunday, Boxing Day, 2010Todays problems are based on SAT Exams. <br /><ol><li> 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.</li><li>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.</li><li>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?</li><li>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), ...</li><li>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?</li><li>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?</li><li>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?</li><li>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?</li><li>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?</li><li>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?</li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-32861567529986277432010-11-21T11:34:00.001-05:002010-11-21T11:37:56.818-05:00Sunday November 21, 2010<ul><li>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.</li><li>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.</li><li> 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)).</li><li>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?</li><li>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.</li><li> 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?</li><li>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?</li><li>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.</li></ul>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-82237037319059724222010-11-14T17:43:00.000-05:002010-11-14T17:43:04.398-05:00November 14, 2010The first one is from Mathematics Olympiad Problems (1996 Vologda). The probability ones are inspired from enrich.maths.org<br /><ul><li>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? </li><li>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!)</li><li>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?</li><li>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.</li><li>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?</li><li>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?</li><li>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?</li></ul>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-7407875629581079132010-11-07T09:06:00.024-05:002010-11-08T14:48:22.836-05:00Sunday, November 7, 2010<ul><li>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.</li><li>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?</li><li>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?</li><li>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.</li><li>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?</li><li>Ad is taller than Rd by 25%. Then, by what percentage Rd is shorter than Ad?</li><li>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? </li><li>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?</li><li>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. <br /></li></ul>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-18200466476113840392010-10-17T14:42:00.003-04:002010-10-17T17:02:26.533-04:00Sunday, October 17, 2010Some more problems based on the NTSE Exam for Grade 8.<br /><ol><li>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?</li><li>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.</li><li>A rectangular gallery has sides in the ratio of 5:2, and its total area is 100 square-mts. What are the side lengths?</li><li>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. </li><li>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. </li><li>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.</li><li>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?</li><li>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. </li><li>Show that the product of any two consecutive even numbers is divisible by 4? Is it divisible by 8?</li><li>What about the product of any three consecutive even numbers? Is it divisible by 16?</li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-66384234735580872252010-09-18T17:25:00.000-04:002010-09-18T17:25:30.579-04:00Saturday, September 18, 2010Today's problems are adapted from the NTSE series of books by Tata McGraw Hill for Class 8th (2008). <br /><ol><li>How many non-overlapping discs of radius 1cms and fit in a disc of radius 2, 4, and 8cms?</li><li>Given a disc D of radius 2 cm lying on a table, whats the maximum number of discs of the same radius you can place (lying on the table) so that each of them is touching D and none of the discs overlap. </li><li>These are classical packing problems: for example how many balls of radius 1cm can fit in a giant ball of radius 4cms? How can we estimate this number? How can we estimate the volume of the gaps? </li><li>The train coming from Toronto to Ottawa, traveling at a speed of 60Km/Hr made me wait for 10 seconds at the signal. How long (in mts) is the train?</li><li>If Ms. Rich's income is 25% more than Mr. Poor, then by what percentage is Mr. Poor's income less than Ms. Rich?</li><li> A door-to-door salesman sells two cookie jars, each for $10.00 - one to Ms. Tough-Bargain and the other to Mr. What-ever. For Ms. Tough-Bargain he make a loss of 10% and for Mr. What-ever he makes a profit of 10%. Did the salesperson made an overall profit or loss?</li><li> Anant's piggy bank has in all change for $33.00. It consists only of quarters, dimes and nickles, in the ratio of 5:3:2. How many coins of each type Anant has?</li><li>We know that in an equilateral triangle, each interior angle is 60 degrees; in a square it is 90 degrees; what do you think it will be in a regular pentagon; regular hexagon; and how we will go about finding it. </li><li>The Rideau River near my department has pretty fast current. Ottawa police does their rescue drill operations once in a while. They go down-stream, and then up-stream. In 10 minutes, the rescuers travel the downstream in their inflatable boats covering the distance of 1 km. Upstream, for the same distance, it takes them twice as long. What is the speed of the current?</li><li>In the school's Terry Fox Run, I came 19th from the top and 235th from the bottom. How many kids participated in the run? </li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-40128684186596797022010-09-06T10:15:00.001-04:002010-09-06T10:16:26.150-04:00September 6. 2010 (Labour Day)We met last two weeks, and did problems from the book "Learning Maths 6B by Norrin Hasim (Singapore Asian Publications)". I highly recommend this book for middle school kids for a variety of very nice problems - I haven't seen other levels of this book - but 6B has lots of nice geometric problems. Today's Problems 1-3 are based on this.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/_tyBV_kd1CXU/TITQRNVNVgI/AAAAAAAAAAg/glUNTybEi3Q/s1600/area-p1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="http://4.bp.blogspot.com/_tyBV_kd1CXU/TITQRNVNVgI/AAAAAAAAAAg/glUNTybEi3Q/s200/area-p1.png" width="200" /></a></div><ol><li>Find the area of the American Football in the square of side 45 cm long, in the above picture. </li><li>1/5th of Anant's mm's are same as that of 1/3rd of Aditya's. If Aditya gives Anant 24 mm's (unlikely though!), Anant will have thrice as many mm's as Aditya. How many they had to begin with?</li><li> A solid iron ball of volume 576 cubic-centimeter is cast into nine identical solid cubes. What are the dimensions of the cube?</li><li>What about the other way around? Suppose you have 16 identical cubes, each having a side length of 2cm long, is cast as a solid ball. What is its radius?</li><li>A string forms an equilateral triangle, whose sides are 3.14cms long. The same string is now reshaped as a circular ring. What is the radius of the ring? </li><li>A circular race track has a inner radius of 56m and outside radius of 63m. What is the total area of the track itself?</li><li>In the above problem, if the 7mts wide track is divided into 7 equally spaced ones- 1mts each, 7 racers need to run the full length of the track once - how should you place them so that each of them cover the same length? </li><li>The difference between the ages of Mr. Young and Ms. Old is 45 years and the ratio of their ages is 3:8. What are the ages of Mr. Young and Ms. Old.</li><li>Divide 25 into two parts, such that 4 times of one part is the same as 6 times the second. </li><li>In a large movie hall, seats are assigned with respect to row and column numbers. In all there are 1296 seats, and the number of rows are the same as columns - how many rows are there?</li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-29545060676750301492010-06-20T10:39:00.000-04:002010-06-20T10:39:55.421-04:00Sunday June 20, 2010This is the last week of the school for the kids, and after this week we will take a break for about 7 weeks - as all of us are travelling to different parts of the world.<br /><br /><ol><li>9 zebras weight same as 4 yaks, 8 yaks weigh same as 15 xantus, 10 xantus weigh same as 27 wolves. How many wolves weigh the same as 4 zebras?</li><li>On a 200m circular race track in your school yard, Bruce and Anant are competing in a race, a long 10KM race. Bruce runs at the speed of 5m/sec and Anant runs at the speed of 4m/sec. How many meters has Anant run when they meet for the first time? Second time? (Both of them are running in the same direction and start at the same time.) </li><li>Aditya and Mark smuggled 72 candies for their trip to the Arrowhead camp. Aditya being scared of getting caught, gives half of his candies to Mark, and since he is still very very scared, he gives in addition 12 more. Mark is thrilled - he has now become the Candy King, he has three times more than what Aditya is left with. How many candies each of Aditya and Mark had to begin with? </li><li>3/4 of 30 is same as 9/10th of what number?</li><li>Radhika buys her favourite sandal for a discount of 20%. Raju, while strolling in Bayshore, spots the same sandal in two different stores. The one is selling it at an additional discount of 20% on what Radhika paid, the other store is selling it at a discount of 37% at the original price. Which deal is better for Raju?</li><li>Two twins, Dharmesh and Prince, have a birthday on Feb 29th. Geeta, his mentor, decided to give Dharmesh $1 on Feb 1, $2 on Feb 2, $3 on Feb 3, ..., $29 on Feb 29th as the B'day gift. Remo, Prince's mentor, decided to give him $20 everyday from Feb 1 to Feb 29th as his B'day gift. Who had a better present?</li><li>Today's geometric construction - we will draw a parallelogram whose sides are 6cm long, angle is 45-degrees. What is its area? </li> </ol><ol> </ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-26026528841431028932010-06-06T10:01:00.000-04:002010-06-06T10:01:23.345-04:00Sunday, June 6, 2010<ol><li>Suppose you have many carpet runners. Their dimension is typically 2ft by 6ft. Your room measure 8ft by 12ft. What kind of interesting patterns you can make to cover your room wall to wall.</li><li>I have tossed a coin 10 times in a row, and so far I haven't got the Heads. (Wow! I am so unlucky!). Does this increases my chance of getting a head in the next toss?</li><li> What's common between golf, notebook and doughnut (or medhu vada).</li><li>Take a <a href="http://en.wikipedia.org/wiki/Canadian_1_dollar_coin">loonie (Canadian 1$ coin) </a>and roll it along its edge on a table. Observe the eye of the loon (the bird/ or the Queen), and trace the curve it follows. What shape is this curve? (Learn More: <a href="http://en.wikipedia.org/wiki/Cycloid">Cycloid</a>) </li><li> Alice and Bob are in two different rooms, and they have no way to communicate with each other as well as they do not trust each other. Think both from Alice and Bobs viewpoint and reason what you will do in the following scenario: Alice gets a $10 bill with the following promise - either she can keep the money or send it to Bob, and if Bob returns the favor then her amount will be doubled (Bob may just walk off). Same is told to Bob, that either he can keep the money which came from Alice or send it back to Alice, and if she returns the favor then the amount which he had will be doubled. They can keep sending back and forth the money and keep doubling! What will you do if you are Alice? or Bob?</li><li> In the news we have heard a lot about the oil spill in the Gulf of Mexico. Here is a simple Math problem related to this. They claim that 26,500 barrels of oil is spilled in a day (1barrel= approx. 160 l). The diameter of the pipe from which the crude oil is oozing out is 20 inches. How many barrels/per day you think they will be spilling out in case they made the diameter of that pipe 24 inches instead of 20 inches.</li><li>An Olympic size swimming pool measures L=50m x W=25m x D=2m. What is the total volume of water it contains? 1 cubic meter requires 1000 liters of water. In terms of the Gulf oil spill, it started 48 days ago - how many pools of oil it is? If the pipe was 24 inches wide then how many swimming pools it would have filled?</li><li>Can you arrange 9 dots on a piece of paper so that there are 10 straight lines, each passing through at least 3 dots.</li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-66626612212799773272010-05-30T10:42:00.001-04:002010-06-05T17:11:40.145-04:00Sunday May 30, 2010<ol><li>Ethiopian Tune ran 10K in 32 minutes and 11.5 seconds. What was her speed in Km/Hr? </li><li> In comparison, in Vancouver 2010 olympics, 10k speed skating record for men was just under 13 minutes. What the speed in km/Hr.</li><li>Usain Bolt ran 100m in 9.58 seconds. What is his speed?</li><li>100m swimming free style record is 46.91 seconds by Cesar Cielo. Whats the speed? </li><li>Niemi save percentage in the goal is .916. If Philly wants to score 5 goals in a game, how many shots, on average, needs to be directed at the Chicago's net?</li><li>In a class, there are 15 young men and 15 young women. When polled, it turns out that each men dates 4 women, and each women dates 3 men. Is it possible? </li><li>You have invited 10 guests for a party. A custard can serves three guests. How many cans you need?</li><li>On a windless Sunday morning, you biked on the Colonel By Drive (which is essentially a straight North-South road) for its entire 10 km length, starting at one end, reaching the other end, and then coming back to the start point. Following Sunday you did exactly the same, but there is a 25 KM/Hr wind blowing from North. Suppose the amount of effort you put in is identical on both the days - will the total time of trip on both the Sundays will be same?</li><li>What music to play when climbing the Eiffel Tower. There are in all 1665 steps to reach the top of the tower (actually we can't since this is an Emergency Exit!). Suppose you can climb at a constant pace, determine how many steps you can climb in a minute - whats the right beat - which is perfect music - and how long will it take?</li><li>According to the Canadian Tire Paint Calculator, a room of dimensions 4m times 4m times 3m requires 5.58 l of paint for the walls and 1.86 lit for the ceilings. Given this information how will you determine how much paint you will require for your room?</li><li>Geometric Construction: Draw a triangle with sides 9cm, 7cm, and 5cm and then determine its area. </li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-75635245220298214552010-05-24T13:26:00.001-04:002010-05-24T13:35:49.656-04:00Monday, May 24, 2010 (Victoria Weekend)The first two problems are motivated from "Our days are numbered" by Brown.<br /><ol><li>Here is a simple game about mind reading: Bob tells Alice to guess two numbers between 0 and 9. Bob asks Alice to perform the following four steps (a) multiply the first number by 5 (b) add 9 to the product (c) double the sum (d) lastly add the second number to the result. Then he asks Alice to tell him the result - and then he subtracts 18 from her answer, and now he knows both the numbers which Alice guessed! Why it works? (For example if Alice guessed 3 and 5, then the steps results in (a) 3x5=15 (b) 15+9=24 (c) 2x24=48 (d) 48+5=53. Bob subtracts 18 from 53 and it equals 53-18= 35 - the two digits which </li><li>Show that the remainder, when a number is divided by 9, is the same as the sum of its digits (if the sum exceeds 9, then keep taking the sum of the digits, till it is below 9). For example, we know that 345/9 results in a remainder of 3. The sum of the digits of 345= 3+4=5=12, which in turn is 1+2= 3, and it is the same as the remainder.</li><li> Continuing on the theme of the angle bisector from the last week, how will you draw two lines which are perpendicular to each other (only compass and ruler).</li><li>Give a line L and point p (p is not on L), how will you find the closest point to p on L geometric construction..</li><li>Given a triangle ABC, how will you draw an incircle of ABC. An incircle is the largest circle completely contained in the triangle. By the way the center of this circle is the meeting point of the angle bisectors.</li><li>What about circumcircle of ABC - that is the smallest circle that completely contains ABC. Its center is on the perpendicular bisectors of the sides of triangle. </li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0tag:blogger.com,1999:blog-6023621475095246539.post-58148365385585836352010-05-16T13:54:00.000-04:002010-05-20T09:41:19.652-04:00May 16, 2010<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/_tyBV_kd1CXU/S_U53aFSrNI/AAAAAAAAAAM/POlItRpwpUc/s1600/circim" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="http://3.bp.blogspot.com/_tyBV_kd1CXU/S_U53aFSrNI/AAAAAAAAAAM/POlItRpwpUc/s200/circim" width="200" /></a></div><ol><li>The four circles of equal radius (say 1cm each) are arranged in such a way that their centers make a perfect square, where each side of the square is 2cm long. What will be the dimension of the little circle that can be placed between all these four circles, such that the little circle just barely touches the four big circles. What is its area compared to the big circles? (Lean More: <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem">Pythagoras Theorem</a>).</li><li>Actually it may be best that you try to draw the above figure using your geometry set. First draw the square, then the four circles, and if your drawing is perfect then the red circle will touch all the other four circles! </li><li>How many numbers are there between 1 and 30, which are not prime, not a multiple of 4 or 5, and are divisble by 2 and 6.</li><li>Consider two points A and B which are at distance 5cm apart on the plane. Rotate A around B at each possible angle (300,270, 180, 90, 45, 30, ...). Consider all the images of A. What shape they form? What happens when A and B are in space - what does the angle of rotation means?</li><li>In the previous problem - say A is a disc and B is a point. Then what shape the union of the images of B form?</li><li>How many of the following statements are true? (a) The number of False statements are 1 (b) The number of False statements are 2 (c) The number of False statements are 3 (d) The number of False statements are 4.</li><li>The sum of the ages of a couple is 77 years. Mr. He is now twice as old as Mrs. She was when he was as old as she is now. What are the current ages of Mr. He and Mrs. She?</li><li>How many pennies are there if in your pocket you have 50 coins making a change for a dollar?</li><li>If 5 is added to 1/3rd of a number, it result in 1/2 of that number. Whats that number?</li><li>How will you bisect an angle between two lines, without actually measuting the angle (you are allowed to use the compass).</li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com1tag:blogger.com,1999:blog-6023621475095246539.post-9927476120122268172010-05-09T09:12:00.000-04:002010-05-10T13:49:34.277-04:00May 9, 2010<ol><li>Five years ago Vic's age was 1/3-rd of the Ash's. Now Vic's age is 17. What is the age of Ash right now?</li><li>The sum of the ages of the Dad-Son pair is 45 years right now. Five years ago the product of their ages was 4 times the age of the dad at that time. What is the current age of the Dad and the Son?<br /></li><li>In a chess tournament each of the six players will play all other players exactly once. How many games will be played in all?</li><li>The KidsPlay store bought 6 teddy's for $10, and sold them as 4 for $10. In all they made $60 profit from the teddy's. How many teddy's did they buy?</li><li>In a quiz, I have 7 times as many correct answers as the wrong ones. In all there were 10 dozen questions in the exam. How many did I get right?</li><li>There are two tall cedar trees in the Stonecrest park. The trees are 40m apart, and they are 30 m tall. Suppose there is a rope running from the top of each tree to the bottom of the other tree, how high above the ground the two ropes will intersect.<br /></li><li>Suppose in the previous question, the trees were 20 m and 30 m tall, how high above the ground will the ropes meet?</li><li>Suppose a leaky tap drips two drops of water every second, each drop is about 2ml. How much water is wasted in one year. Suppose you require 2 bucket full of water every day (approx. 40 l), compare that with the amount wasted water. <br /></li></ol>Anil Maheshwarihttp://www.blogger.com/profile/10627754080035687021noreply@blogger.com0