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"Cool" inductive proofs
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Chris Smith
science forum beginner


Joined: 02 Apr 2005
Posts: 11

PostPosted: Wed Jul 19, 2006 5:37 am    Post subject: "Cool" inductive proofs Reply with quote

I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

Thanks,

--
Chris Smith
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Wed Jul 19, 2006 6:01 am    Post subject: Re: "Cool" inductive proofs Reply with quote

On Tue, 18 Jul 2006, Chris Smith wrote:

Quote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

1. Are simple enough to provoke an average 14 year old to think.


Quote:
2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

1 + 2 + 3 +..+ n-1 + n = n(n + 1)/2


Quote:
3. Involve minimal or no algebraic manipulation.

3. Involve exercising simple algebraic notions.


Quote:
The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

Then make the audience suitable for the problems.


Instead of talking down to our youth, of continuing their expectation that
it all comes easy, provoke them to think, stimulate them to learn,
inspire them to enjoy and use their mental skills.
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pomerado@hotmail.com
science forum beginner


Joined: 15 Jun 2006
Posts: 15

PostPosted: Wed Jul 19, 2006 6:10 am    Post subject: Re: "Cool" inductive proofs Reply with quote

Chris Smith wrote:
Quote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

The difference between squares of two successive integers is the sum of
the integers.
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Wed Jul 19, 2006 6:18 am    Post subject: Re: "Cool" inductive proofs Reply with quote

In article <1153289429.086849.56110@75g2000cwc.googlegroups.com>,
"Richard Henry" <pomerado@hotmail.com> wrote:

Quote:
Chris Smith wrote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

The difference between squares of two successive integers is the sum of
the integers.

That scarcely needs induction: (x+1)^2 - x^2 = x + (x+1) is trivial
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Chris Smith
science forum beginner


Joined: 02 Apr 2005
Posts: 11

PostPosted: Wed Jul 19, 2006 6:28 am    Post subject: Re: "Cool" inductive proofs Reply with quote

William Elliot <marsh@hevanet.remove.com> wrote:
Quote:
Then make the audience suitable for the problems.

Instead of talking down to our youth, of continuing their expectation that
it all comes easy, provoke them to think, stimulate them to learn,
inspire them to enjoy and use their mental skills.

Your mind-reading skills are extraordinary, as you seem to know enough
about the situation to make these kinds of judgements without having
been told anything. I appreciate your confidence that I'll be able to
teach algebra to a group of kids half of whom may not have seen it
before in about 45 minutes per day for three days. I'm even more
honored that you think I'd be able to accomplish that while also
providing an interesting experience for the remaining half of the group
that already knows algebra. However, I don't think that's a realistic
appraisal of my skills.

Therefore, I'm still looking for the same thing I originally asked for.

--
Chris Smith
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Gerry Myerson
science forum Guru


Joined: 28 Apr 2005
Posts: 871

PostPosted: Wed Jul 19, 2006 6:52 am    Post subject: Re: "Cool" inductive proofs Reply with quote

In article <MPG.1f2772f07de96a59896a3@news.altopia.net>,
Chris Smith <cdsmith@twu.net> wrote:

Quote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

In a round-robin tournament, if there's any cycle (that is,
any case where a beats b, b beats c, ..., y beats z, and z beats a),
then there's a cycle involving just 3 entrants.

You can make up any amount of postage 18 cents or greater
using just 4-cent and 7-cent stamps.

(1 + x)^n > 1 + nx for x real, x > -1 (except x = 0), n integer,
n > 1.

--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
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Ken Oliver
science forum beginner


Joined: 15 May 2005
Posts: 27

PostPosted: Wed Jul 19, 2006 9:07 am    Post subject: Re: "Cool" inductive proofs Reply with quote

"Chris Smith" <cdsmith@twu.net> wrote in message
news:MPG.1f2772f07de96a59896a3@news.altopia.net...
Quote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

Thanks,

--
Chris Smith

The sum of the interior angles of an n-sided convex polygon is (n-2)*180
degrees. (Given that the sum of the interior angles of a triangle is 180.)
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Richard Tobin
science forum Guru Wannabe


Joined: 02 May 2005
Posts: 165

PostPosted: Wed Jul 19, 2006 9:13 am    Post subject: Re: "Cool" inductive proofs Reply with quote

In article <MPG.1f2772f07de96a59896a3@news.altopia.net>,
Chris Smith <cdsmith@twu.net> wrote:

Quote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

Towers of Hanoi: that you can solve it in 2^n - 1 moves, and that this
is the shortest solution.

-- Richard
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Dirk Van de moortel
science forum Guru


Joined: 01 May 2005
Posts: 3019

PostPosted: Wed Jul 19, 2006 9:19 am    Post subject: Re: "Cool" inductive proofs Reply with quote

"Chris Smith" <cdsmith@twu.net> wrote in message news:MPG.1f2772f07de96a59896a3@news.altopia.net...
Quote:
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The one really good example I've found so far is the ability to tile an
NxN grid with one square removed with L-shaped pieces, provided N is a
power of two. Are there other good examples along these same lines?
Unfortunately, most introductions to induction that I've found seem to
be field-specific, or prove closed forms for members of a series; these
aren't suitable for the audience I've got.

That's a nice example with almost no algebra.

The best example with limited algebra is of course
1+2+...+N = 1/2 N (N+1), but if that is still too much
algebra, try this one:
How many ways to line up a group of N 14-year-olds?
One 14-year-old ==> One way.
a group of N 14-year-olds ==> for each of the possible
line-ups of a group of one less 14-year-olds, there are N
places where you can put the next 14-year-old.
They can participate in the demonstration.

Dirk Vdm
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jmfbahciv@aol.com
science forum Guru Wannabe


Joined: 12 Sep 2005
Posts: 297

PostPosted: Wed Jul 19, 2006 9:56 am    Post subject: Re: "Cool" inductive proofs Reply with quote

In article <MPG.1f277f10a1129c289896a4@news.altopia.net>,
Chris Smith <cdsmith@twu.net> wrote:
Quote:
William Elliot <marsh@hevanet.remove.com> wrote:
clap, clap


Quote:
Therefore, I'm still looking for the same thing I originally asked for.

I can't help you but I do have a suggestion. Collect the examples
that do work and make a booklet. It may only be a little extra
work when you write up the problems for your kids.

Would an exercise in ratios help?

/BAH
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jmfbahciv@aol.com
science forum Guru Wannabe


Joined: 12 Sep 2005
Posts: 297

PostPosted: Wed Jul 19, 2006 11:16 am    Post subject: Re: "Cool" inductive proofs Reply with quote

In article <qJovg.545436$T03.13126712@phobos.telenet-ops.be>,
"Dirk Van de moortel" <dirkvandemoortel@ThankS-NO-SperM.hotmail.com> wrote:
Quote:

jmfbahciv@aol.com> wrote in message
news:e9kvks$8qk_001@s997.apx1.sbo.ma.dialup.rcn.com...
In article <MPG.1f277f10a1129c289896a4@news.altopia.net>,
Chris Smith <cdsmith@twu.net> wrote:
William Elliot <marsh@hevanet.remove.com> wrote:
clap, clap

Therefore, I'm still looking for the same thing I originally asked for.

I can't help you but I do have a suggestion. Collect the examples
that do work and make a booklet. It may only be a little extra
work when you write up the problems for your kids.

Would an exercise in ratios help?

You mean like in
(n-1) n / 2 + n = ... = n (n+1) / 2 ?
Devious Wink

Beatstheshitoutame. It was something that popped into my head.
I can't remember which came first: I either used ratios
to think about induction or induction to think about ratios
which I was learning about one of them. :-)

I figured that, if my idea was full of it, the OP would
simply ignore me.

/BAH
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Dirk Van de moortel
science forum Guru


Joined: 01 May 2005
Posts: 3019

PostPosted: Wed Jul 19, 2006 11:30 am    Post subject: Re: "Cool" inductive proofs Reply with quote

<jmfbahciv@aol.com> wrote in message news:e9kvks$8qk_001@s997.apx1.sbo.ma.dialup.rcn.com...
Quote:
In article <MPG.1f277f10a1129c289896a4@news.altopia.net>,
Chris Smith <cdsmith@twu.net> wrote:
William Elliot <marsh@hevanet.remove.com> wrote:
clap, clap

Therefore, I'm still looking for the same thing I originally asked for.

I can't help you but I do have a suggestion. Collect the examples
that do work and make a booklet. It may only be a little extra
work when you write up the problems for your kids.

Would an exercise in ratios help?

You mean like in
(n-1) n / 2 + n = ... = n (n+1) / 2 ?
Devious ;-)

Dirk Vdm
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Wed Jul 19, 2006 11:41 am    Post subject: Re: "Cool" inductive proofs Reply with quote

On Wed, 19 Jul 2006, Dirk Van de moortel wrote:
Quote:
"Chris Smith" <cdsmith@twu.net> wrote in message news:MPG.1f2772f07de96a59896a3@news.altopia.net...
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The best example with limited algebra is of course
1+2+...+N = 1/2 N (N+1), but if that is still too much

What 14 year old student did that without induction, Gauss?
1 + 2 +..+ n-1 + n
n + n-1 +..+ 2 + 1
------------------
n+1 + n+1 +..+ n+1

n(n+1)/2
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Dirk Van de moortel
science forum Guru


Joined: 01 May 2005
Posts: 3019

PostPosted: Wed Jul 19, 2006 11:46 am    Post subject: Re: "Cool" inductive proofs Reply with quote

"William Elliot" <marsh@hevanet.remove.com> wrote in message news:Pine.BSI.4.58.0607190436180.12173@vista.hevanet.com...
Quote:
On Wed, 19 Jul 2006, Dirk Van de moortel wrote:
"Chris Smith" <cdsmith@twu.net> wrote in message news:MPG.1f2772f07de96a59896a3@news.altopia.net...
I am looking for examples of inductive proofs that:

1. Are simple enough to be understood by an average 14-year-old.

2. Prove theorems that are easily *understood* by the average teenager,
but won't be too obvious so that the need for a proof is clear.

3. Involve minimal or no algebraic manipulation.

The best example with limited algebra is of course
1+2+...+N = 1/2 N (N+1), but if that is still too much

What 14 year old student did that without induction, Gauss?
1 + 2 +..+ n-1 + n
n + n-1 +..+ 2 + 1
------------------
n+1 + n+1 +..+ n+1

n(n+1)/2

"Ligget Sie".
I read somewhere that he was 6 or 7.
Alas, they don't produce Gausses in large quantities ;-)

Dirk Vdm
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Wed Jul 19, 2006 11:52 am    Post subject: [] "Cool" inductive proofs Reply with quote

On Wed, 19 Jul 2006, Chris Smith wrote:

Quote:
William Elliot <marsh@hevanet.remove.com> wrote:
Then make the audience suitable for the problems.

Instead of talking down to our youth, of continuing their expectation that
it all comes easy, provoke them to think, stimulate them to learn,
inspire them to enjoy and use their mental skills.

Your mind-reading skills are extraordinary, as you seem to know enough
about the situation to make these kinds of judgements without having

Thanks for the compliment. My psychic abilities have been tutored by
students unwilling to make mathematically coherent statements.

Quote:
been told anything. I appreciate your confidence that I'll be able to
teach algebra to a group of kids half of whom may not have seen it
before in about 45 minutes per day for three days. I'm even more
honored that you think I'd be able to accomplish that while also
providing an interesting experience for the remaining half of the group
that already knows algebra. However, I don't think that's a realistic
appraisal of my skills.

Where do I guide a group of people, half of whom are in shape and half of

whom wear out in a mile, for a hike that would be a pleasant go nowhere
stroll for the tender footed and a fantastic place to see for those who
keep in shape? Would I not ask for an assistant to take one group around
the picnic tables so the other group could see the waterfall?

No, I'd ask an assistant to take one group to the base of the water fall
and return to the picnic tables to rest up while the rest went on to the
top of the water fall and back.

Quote:
Therefore, I'm still looking for the same thing I originally asked for.

I'm wanting schools that don't dampen kids' enthusiasm for learning.
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