set1
Fibonacci number and ratio generator
e approximator
Set 2
4/26/00
- Prove the Pythagorean theorem by any means. Use geometry, graphs or trigonometry
as needed.
- In a water molecule (two hydrogen atoms and one oxygen atom), the three atoms
form a bent line with an angle of 109° between the central oxygen atom and two
hydrogen atoms. Prove this 109° angle by geometry and find it to 6 significant
digits. Hint: the atom will fit inside a cube when the oxygen atom is
in the center of the "bottom" face and the hydrogen atoms are on the "top" face in
opposite upper corners.
- A icosahedron is a 3-dimensional object with 12 triagonal faces of the
same size. Find the number of vertices and the number of edges in a icosahedron.
If the length of one side of a triangle is 1cm,
find the distance from the center of one triangle to the center of the opposite
triangle. Link to
http://mathworld.wolfram.com/Icosahedron.html for a very good explanation on
the icosahedron, dodecahedron and triambic pentahedron and many many more.
- The Euler-Mascheroni constant, given the symbol gamma, can be
approximated by the following sum: (the "lg n" is the base 2 logarithm of n).
Approximate gamma the hard way by doing a few of the sums and write a theoretical program
loop (assuming that "lg" is a valid function) that could solve gamma to a large number of
decimal places (or write a working program if you know how). The L-shaped braces around
the "lg n" function means to "floor", or round down the "lg n" result, e.g. floor[2.431]
= 2. See the gamma approximator.
- Two cars start 50 miles apart travelling toward each other on a straight
road. Car 1 is going 30 miles per hour, while car 2 is going X miles per hour. Car
2 starts travelling 15 minutes after car 1. Find X such that the two cars meet 18
miles from where car 2 started. How long did it take for the cars to meet?
- If 4 clowns can juggle 14 balls (each juggles independently), and one
clown quits, what is the probability that the remaining clowns can juggle 11 balls?
- Invent a technique or write a simple program to approximate the base 2
logarithm of a number.
- A triangular number is created by making a triangle out of a certain number
of objects, like bowling pins. A few numbers in this series are 1, 3, 6, 10, ... Find
the 100th and 101st elements in the series and find their difference and ratio. What
numbers do their difference and ratio approach? What can you say about the relation
of the distance to that particular element in the series? Also try the
Triangle Approximator program.
- Show that
 given that
the subfactorial, "! n", is as defined below:
This is the only number which is equal to the subfactorial of its digits.
- Find the ratio between adjacent squares formed by using adjacent numbers
in the Fibonacci series as the edges, that is, the first few squares are 0*1 = 0,
1*1 = 1, 1*2 = 2, 2*3 = 6, 3*5 = 15.... and the ratios are 0/1 = 0, 1/2 = 0.5,
2/6 = 0.33333, 6/15 = 0.4 ..... what can you say about the number this series approaches?
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