# Skid Patch Theorem

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**Skid Patch Theorem**

Theorem:

(2) is generally not known and disproves several conjectures seen recently.

Proof of (1):

Let(1) is already well-known. It is proved here for completeness.a/bbe the reduced gear ratio (that is,aandbare integers with no common divisors other than 1). Then,

(1) With single-sided skidding, there arebskid patches, and

(2) Ambidexterous skidding doubles the number of skid patches if and only ifais odd.

(2) is generally not known and disproves several conjectures seen recently.

Proof of (1):

Turning the pedals through one revolution turns the wheel throughProof of (2):a/brevolutions. Turning the pedals throughbrevolutions turns the wheel throughb* (a/b) =arevolutions. That is, afterbpedal revolutions, the wheel is returned to the same position it was originally (sinceais an integer). So there must be no more thanbskid patches, since the same cycle ofbwheel positions will be repeated through everybpedal revolutions.

Now suppose that two of the intermediate wheel positions were the same, say, afteriandjpedal revolutions (0 <=i<j<b). Thenj-ipedal revolultions also returns the wheel to its original position, so (j-i) *a/bis an integer. Thusbmust evenly divide (j-i) *a/ b. However,aandbhave no common divisors, sobmust evenly dividej-i. Butj-iis less than b, so this cannot happen. Therefore, allbof the intermediate wheel positions (after 0, 1, 2, ..., andb-1pedal revolutions) are different. So there must be no fewer thanbskid patches.

There are no more thanbskid patches and there are no fewer thanbskid patches, so there must be exactlybskid patches.

As above, turning the pedals through one revolution turns the rear wheel througha/brevolutions. Turning the pedals through one half-revolution turns the rear wheel through half as many revolutions. So the number of skid patches with ambidexterous skidding should be the same as that with single-sided skidding on a gear ratio half as large. Now to apply (1) to this situation, we need to know how 1/2 *a/breduces as an integer ratio. This depends on whetherais even or odd. Ifais even, (a/2) /bis the reduced ratio, so there arebskid patches, as in the single-sided case. Ifais odd,a/ (2b) is the reduced ratio, so there are 2bskid patches.

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**2**antisocialite

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yup. we were just talking about this earlier in the week. i was thinking that a AND b need to be odd, but i agree with you that it's just a (not that it matters what i agree with, nice proof )

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**4**Velorution

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omg what did u get in differential and calc 3. don't tell me, an A. i'm going to go over ur theorem with the math club tonight. hopefully ur right.

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This new learning amazes me, Sir Bedevere. Explain again how sheep's bladders may be employed to prevent earthquakes.

__________________

This is Africa, 1943. War spits out its violence overhead and the sandy graveyard swallows it up. Her name is King Nine, B-25, medium bomber, Twelfth Air Force. On a hot, still morning she took off from Tunisia to bomb the southern tip of Italy. An errant piece of flak tore a hole in a wing tank and, like a wounded bird, this is where she landed, not to return on this day, or any other day.

This is Africa, 1943. War spits out its violence overhead and the sandy graveyard swallows it up. Her name is King Nine, B-25, medium bomber, Twelfth Air Force. On a hot, still morning she took off from Tunisia to bomb the southern tip of Italy. An errant piece of flak tore a hole in a wing tank and, like a wounded bird, this is where she landed, not to return on this day, or any other day.

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Jesus, I was trying to blow off my number theory homework not do extra curricular work.

Anyways, nice proof.

Anyways, nice proof.

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I nominate this thread for "Best Title of the Week"

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**10**dig dig dig

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just rotate your tire a few degress every so often. prob. = solved

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Originally Posted by

**Moximitre**just rotate your tire a few degress every so often. prob. = solved

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**12**Al

Maybe its time to update my SP chart and put the ones to double for ambi in italics or somesuch. Last update was may-05

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I've added a "skid patch analysis" function to rabbit showing the conclusion of Fraction's SPT is valid.

Check it out at http://software.bareknucklebrigade.com if interested.

Check it out at http://software.bareknucklebrigade.com if interested.

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theres a book called http://en.wikipedia.org/wiki/Cryptonomicon it has a cool discription of how code machines work n it uses a bike to describe it.

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Originally Posted by

**thatcher**theres a book called http://en.wikipedia.org/wiki/Cryptonomicon it has a cool discription of how code machines work n it uses a bike to describe it.

This was when I was a math major but yet to get into bikes.

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ok. I am having a little trouble understanding exactly what a skid patch is. I'll take a guess and say every rotation of the pedals you lock up the wheels once or twice (single-sided or ambidextrous)?

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**20**antisocialite

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ask and ye shall be linked:

http://sheldonbrown.com/fixed.html#skid

edit: if the theorem above is correct, and it looks like it is, the sentence should read:

If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.

If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.

to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong

http://sheldonbrown.com/fixed.html#skid

edit: if the theorem above is correct, and it looks like it is, the sentence should read:

If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.

If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.

to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong

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**21**jack of one or two trades

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Originally Posted by

**shogun17**ok. I am having a little trouble understanding exactly what a skid patch is.

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**22**Sheldon Brown's posse

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ah thankyou. That explains a lot.

get a brake.

get a brake.

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**23**oldsprinter

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Originally Posted by

**dirtyphotons**ask and ye shall be linked:

http://sheldonbrown.com/fixed.html#skid

edit: if the theorem above is correct, and it looks like it is, the sentence should read:

If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.

If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.

to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong

http://sheldonbrown.com/fixed.html#skid

edit: if the theorem above is correct, and it looks like it is, the sentence should read:

If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.

If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.

to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong

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**24**Loners

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Originally Posted by

**nexus6**You just run a 17t cog, problem solved!

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Originally Posted by

**dirtyphotons**http://sheldonbrown.com/fixed.html#skid

edit: if the theorem above is correct, and it looks like it is, the sentence should read:

If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.

If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.

to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong

Originally Posted by

**fraction**Let a / b be the reduced gear ratio (that is, a and b are integers with no common divisors other than 1). Then,

...

(2) Ambidexterous skidding doubles the number of skid patches if and only if a is odd.

...

(2) Ambidexterous skidding doubles the number of skid patches if and only if a is odd.