Thursday, November 12, 2009
Wednesday, November 11, 2009
Thursday, May 7, 2009
An interesting pattern
Find a pattern in the following
1 squared is 1; 11 squared is 121; A difference of 120
2 squared is 4; 12 squared is 144; A difference of 140
3 squared is 9; 13 squared is 169; A difference of 160
4 squared is 16; 14 squared is 196; A difference of 180
5 squared is 25; 15 squared is 225; A difference of 200
Isn't interesting(For those who have a passion for math)?
This goes on ForeverThen, for numbers ending in the number 1(1,11,21,31...)
1 squared is 1
11 squared is 121
21 squared is 441
31 squared is 961
41 squared is 1681
51 squared is 2601
61 squared is 3721
71 squared is 5041
Now obviously, the ones digit is always 1. the Tens digit goes up by 2s, and the hundreds digit goes up 1,3,5,7,0,1,3
The reason behind this is
we are figuring out:(x + 10)² - x²
Expanding out you have
x² + 20x + 100 - x²= 100 + 20x
This matches exactly with your results which is 120, 140, 160, etc.
For your second sequence.Your numbers are of the form:10x + 1
Squaring you have:(10x + 1)²= 100x² + 20x + 1
The ones digit is always 1 because of the +1.The number goes up by 20 each time because of the +20x, which means the tens digit goes up by 2.The amount above 100 goes up by a square (plus the carry from the tens digit):0, 1, 4, 9, 16, 25(+1), 36(+1), 49(+1)the difference would be odd numbers, but it bumps up by the carry on every 5 numbers.1, 3, 5, 7, 9(+1)=10-->0, 1, 3, 5, 7, 9(+1)=10-->0
1 squared is 1; 11 squared is 121; A difference of 120
2 squared is 4; 12 squared is 144; A difference of 140
3 squared is 9; 13 squared is 169; A difference of 160
4 squared is 16; 14 squared is 196; A difference of 180
5 squared is 25; 15 squared is 225; A difference of 200
Isn't interesting(For those who have a passion for math)?
This goes on ForeverThen, for numbers ending in the number 1(1,11,21,31...)
1 squared is 1
11 squared is 121
21 squared is 441
31 squared is 961
41 squared is 1681
51 squared is 2601
61 squared is 3721
71 squared is 5041
Now obviously, the ones digit is always 1. the Tens digit goes up by 2s, and the hundreds digit goes up 1,3,5,7,0,1,3
The reason behind this is
we are figuring out:(x + 10)² - x²
Expanding out you have
x² + 20x + 100 - x²= 100 + 20x
This matches exactly with your results which is 120, 140, 160, etc.
For your second sequence.Your numbers are of the form:10x + 1
Squaring you have:(10x + 1)²= 100x² + 20x + 1
The ones digit is always 1 because of the +1.The number goes up by 20 each time because of the +20x, which means the tens digit goes up by 2.The amount above 100 goes up by a square (plus the carry from the tens digit):0, 1, 4, 9, 16, 25(+1), 36(+1), 49(+1)the difference would be odd numbers, but it bumps up by the carry on every 5 numbers.1, 3, 5, 7, 9(+1)=10-->0, 1, 3, 5, 7, 9(+1)=10-->0
Wednesday, May 6, 2009
Efficiency of packing
In this project you can calculate the efficieny of packing spherical,circular or cylindrical object in a box with given dimensions in two ways.
1. Square packing. In this arrange spheres in square manner i.e. the centres of adjoining spheres must make a square then side of square is 2r if radius of the sphere is r. Calculate the volume of the box and volume occupied by the balls
efficiency = volume ocuupied by object * 100 %
---------------------
volume of the box
2. Hexagonal packing. In this you can arrange spheres in the hexagonal manneri.e. the centres of adjoining spheres make hexagon and calculate its efficiency.
Observe which packing is efficient.
Also you can relate various examples in daily life to ensure which packing is efficient.
Transformation using co-ordinate geometry
In this project you can choose any shape like triangle, square, parallelogram and find its area using concept of co-ordinate geometry. Then apply the transformation like translation,rotation ,reflection i.e. mirror image of object along axis and observe the effect on area of the shape. The area remains invariant under all the circumstances.
Thursday, April 23, 2009
Penrose tiles
Penrose tiles
There are a number of regular shapes which "tile the plane", just by using large numbers of the same tile. Squares and rectangles, for example, can be laid down in a neat matching pattern that will go on forever. Hexagons, the sort that we see on bathroom floors, are another example of a tile which will cover the whole of an infinite plane, ignoring the raggedy bits around the edges (because infinite planes don't have edges!). NOTE: these diagrams are incomplete, so use the ones below, which have circle segments in them.
Other shapes do the same thing: right-angled triangles can be put together to make rectangles and squares, while equilateral triangles can be assembled into hexagons, and so on. Then there are sets of tiles that work together. Octagons cannot tile the plane by themselves, but if you add in some small squares, you have a complementary set of two shapes which will tile the plane.
All of these tiles and tile sets produce a periodic tiling pattern, where if you travel long enough across the plane, you will find the same pattern repeating itself. The main thing about Penrose tiles is that they seem to be able to tile the plane in an aperiodic way. That is, you can entirely cover an infinite surface, always moving outwards, but without producing any repeating patterns.
Penrose tiles are named after their inventor, Sir Roger Penrose, and they typically show a sort of five-fold symmetry, which derives from the fact that the angles are all multiples of 36 degrees. In the tile sets shown here, the angles are 216, 36, 36 and 72 degrees in the top tile, and 144, 72, 72 and 72 degrees in the lower tile.
In another tile set, the angles are 144, 144, 36 and 36 degrees for one tile, and 108, 108, 72 and 72 degrees for the other. Your task is to make a tile set in large numbers: at least 30 or 40 of each of the two tiles in that set, and to see if you can fit them all together.
There are a number of regular shapes which "tile the plane", just by using large numbers of the same tile. Squares and rectangles, for example, can be laid down in a neat matching pattern that will go on forever. Hexagons, the sort that we see on bathroom floors, are another example of a tile which will cover the whole of an infinite plane, ignoring the raggedy bits around the edges (because infinite planes don't have edges!). NOTE: these diagrams are incomplete, so use the ones below, which have circle segments in them.
Other shapes do the same thing: right-angled triangles can be put together to make rectangles and squares, while equilateral triangles can be assembled into hexagons, and so on. Then there are sets of tiles that work together. Octagons cannot tile the plane by themselves, but if you add in some small squares, you have a complementary set of two shapes which will tile the plane.
All of these tiles and tile sets produce a periodic tiling pattern, where if you travel long enough across the plane, you will find the same pattern repeating itself. The main thing about Penrose tiles is that they seem to be able to tile the plane in an aperiodic way. That is, you can entirely cover an infinite surface, always moving outwards, but without producing any repeating patterns.
Penrose tiles are named after their inventor, Sir Roger Penrose, and they typically show a sort of five-fold symmetry, which derives from the fact that the angles are all multiples of 36 degrees. In the tile sets shown here, the angles are 216, 36, 36 and 72 degrees in the top tile, and 144, 72, 72 and 72 degrees in the lower tile.
In another tile set, the angles are 144, 144, 36 and 36 degrees for one tile, and 108, 108, 72 and 72 degrees for the other. Your task is to make a tile set in large numbers: at least 30 or 40 of each of the two tiles in that set, and to see if you can fit them all together.
Saturday, March 21, 2009
Mind Reading cards
You can create mind reading cards and make befool others that you can read their minds.
then make few other cards involving few symbols according to the logic
ask the person to tell you in which card the symbol is there and then answer them.
there are few steps to do that
1. Remind the equation 2^n - 1 and follow point where n is the no. of boxes or cards we want to design
2. Solve this equation assuming any value of n.
E.g. Make a card and write few symbols on it (remember the total no is 2^n - 1)
E.g. Make a card and write few symbols on it (remember the total no is 2^n - 1)
then make few other cards involving few symbols according to the logic
ask the person to tell you in which card the symbol is there and then answer them.
This project was submitted by Sahiba Mittal of XG
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