Some Examples of Mathematical Induction

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Some Examples of Mathematical Induction
Example 1. Number of intersecting points
Given n lines on a plane. Suppose
(i) No two lines are parallel,
(ii) No three lines are concurrent.
Question How many intersecting points ?
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Some Examples of Mathematical Induction
When n 4, No. of intersecting points 6
When n 3, No. of intersecting points 3
When n 2, No. of intersecting points 1
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Some Examples of Mathematical Induction
Question How many intersecting points ?
From above, we have a table
n 2
1
(2)(1)/2
(3)(2)/2
n 3
3
(4)(3)/2
n 4
6
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Some Examples of Mathematical Induction
Guess For n lines,
Question How to prove it ?
Answer Try Mathematical Induction
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Some Examples of Mathematical Induction
Idea When n 2, it is true. (direct checking)
Assume that it is true when n k.That is, there
are k(k-1)/2 intersecting points.
When n k1, we have (k1) lines.You may view
them as a system of k lines and one extra line is
added on it.
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Some Examples of Mathematical Induction
By induction hypothesis, the k lines have k
(k-1)/2 intersecting points.
Since (i) No two lines are parallel, and (ii)
No three lines are concurrent
The last line intersects the k lines at k
distinct points.
Totally we have k(k-1)/2k intersecting
points.That is, (k1)(k)/2 intersecting points.
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Some Examples of Mathematical Induction
By the principle of mathematical induction, the
proposition is true for n 2, 3, 4, ...
That is, for n given lines,
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Some Examples of Mathematical Induction
Example 2. Towers of Hanoi
Towers of Hanoi is a classical game.
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Some Examples of Mathematical Induction
You have a small collection of disks and three
posts onto which you can put the disks. The disks
all start on the leftmost post, and you want to
move them to the rightmost post. The rules are as
follows
(i) Never put a disk on top of a smaller one.
(ii) The middle post is for intermediate storage.
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Some Examples of Mathematical Induction
Question What is the least number of steps for
n discs ?
Try it !
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Some Examples of Mathematical Induction
From above, we have a table
n 2
3
22 - 1
23 - 1
7
n 3
24 - 1
n 4
15
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Some Examples of Mathematical Induction
Some Examples of Mathematical Induction
Guess For n discs,
Again, how to prove it ?
Answer Try Mathematical Induction (again !)
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Some Examples of Mathematical Induction
Idea When n 2, it is true. (direct checking)
Assume that it is true when n k.That is, we
can move all the k discs to the leftmost post in
2k1 steps.
When n k1, we have (k1) discs.We can first
move the upper k discs to the middle post.It
requires 2k1 steps by induction hypothesis.
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Some Examples of Mathematical Induction
Then we move the bottom (k1)th discs to the
leftmost post. This requires 1 step.
Finally, we move the k discs in the middle post
to the leftmost post. Again it requires 2k1
steps by induction hypothesis.
So total number of steps for k1 discs
(2k1) 1 (2k1) 2k 2k 1 2 (2k) 1
2k1 1
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Some Examples of Mathematical Induction
By the principle of mathematical induction, the
proposition is true for n 2, 3, 4, ...
That is, for n discs,
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Some Examples of Mathematical Induction
There is a legend about the Towers of Hanoi At
the beginning of time the priests in a temple
were given a stack of 64 gold disks. Their
assignment was to transfer the 64 disks from the
leftmost post to the rightmost post. The priests
worked very efficiently, day and night. When they
finished their work, the myth said, the temple
would crumble into dust and the world would
vanish.
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Some Examples of Mathematical Induction
By the above, The number of separate transfers of
single disks the priests must make to transfer
the tower is 2641 18,446,744,073,709,55
1,615 moves!
If the priests worked day and night, making one
move every second it would take slightly more
than 580 billion years (i.e. 580,000,000,000
years) to accomplish the job!
It is estimated that the sun will run out of
hydrogen fuel and this will result in the total
destruction of the Earth after 4.5 billion years.
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Some Examples of Mathematical Induction
You may download the game of Towers of Hanoifrom
the following websites
http//sac-ftp.gratex.sk/educult31.html , or
http//www.lhs.berkeley.edu/java/tower/towerhist
ory.html
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The End