PowerPoint Presentation
Shooting Pool
By: Alvaro Francisco ManuelWestern Oregon University
After the intro, Ill show this animation. 2
Ask what they noticed. Number of bounces? Angle shoot at? Size
of pool table?27SummaryWe have a (3,4) pool table.Starting position
was at the lower left corner.Five bounces before landing in the
lower right corner pocket. Problem StatementOur task is to figure
out a general formula to predict the number of bounces before
landing in a corner pocket for any size pool tableAs well as,
predicting which corner pocket.
Assumption
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We are going to assume that we are just using the cue ball, cue
stick, and pool table. We are assume that the cue ball travels at
the same velocity. Hit at a 45 degree angle.Four corner pockets and
no side pockets.
30NotationWe will refer to our pool table with two dimensions as
(n,m)n corresponds to the vertical dimension (y-axis) and m
corresponds to the horizontal dimension (x-axis)Lower Left: LLLower
Right: LRUpper Left: ULUpper Right: URT(n,m) is defined to be the
size of pool table.B(n,m) is defined to be the number of bounces
P(n,m) is defined to be the corner pocket the ball lands
in.gcd(n,m) is defined to be the Greatest Common Divisor of n and
m.lcm(n,m) is defined to be the Least Common Multiple of n and
m.
Q: Do I need lcm? no.31Notation43For example, we have T(3,4).The
starting position is at the LL corner pocket. B(3,4) = 5. P(3,4) =
LR (Lower Right)My strategyThe plan was to work on many examples
starting with small table size.TablesDimensions# of
BouncesLocation1x10UR1x21LR1x32UR1x43LR1x54UR1x65LR2x11UL2x20UR2x33UL2x41LR2x55UL2x62UR2x77UL2x83LR2x99UL2x104UR3x12UR3x23LR3x30UR3x45LR3x56UR3x61LR3x78UR3x89LR3x92UR3x1011LR4x13UL4x21UL4x35UL4x40UR4x57UL4x63UL4x79ULDimensions#
of
BouncesLocation5x47LR5x50UR5x69LR5x710UR6x711UL6x85LR6x93UL6x106UR6x1115UL7x813LR7x914UR7x1015LR8x23UL8x39UL8x410UL8x511UL9x411LR9x512UR9x63LR9x714UR10x66UR10x715UL10x87LR10x917UL11x312UR11x413LR11x514UR11x615LRPatterns
I foundAny T(n m) will have the same number of bounces as the
reverse table i.e. T(m n). For example, look at T(4,5 ) and T(5,4),
they have the same number of bounces. If you rotate a T(4, 5) 90 we
get a T(5,4) so clearly the number of bounces does not change, but
the corner pockets is now in a different relative position. The
corner pockets will be opposites of each other. For example, P(4,5)
= UL and P(5,4) = LR.Look at T(1,2) , T(2,4), and T(4,8), and
notice B(1,2) = B(2,4) = B(4,8) = 3.In general, it seemed that
T(n,m) = T(kn,km)FormulasTheorem 5.1 If the gcd(n,m)= k, then
B(n,m) = B (nk , mk) and P(n,m) = P (nk,mk).Theorem 5.2 If the
gcd(n,m) = 1 then B(n,m) = n + m 2.Theorem 5.3 Let T(n,m) be a
given table with gcd(n, m) = 1. Then P(n,m) = UR n and m odd LR n
odd and m even UL n even and m oddExamplesSolve a T(10,11),
T(15,20), and T(448,320).ExamplesT(10,11) gcd(10,11) = 1. We use
our Them 5.1 B(10,11) = 10+11-2 = 19. Since 10 is even and 11 is
odd, then P(10,11) = ULT(15,20)gcd(15,20) = 5, we use Thm. 5.1 to
get T(3,4). Thm. 5.1/.2 B(3,4)= B(15,20)=5Since 3 is odd and 4 is
even then P(3,4) = P(15,20) = LLT(448,320)gcd(448,320)= 64. By Thm
5.1 we get T(7,5). By Thm. 5.1/.2 B(7,5)=B(448,320)=10Since both 7
and 5 are odd then P(7,5)=P(448,320)= UR.
Any questionsOur starting position is always going to at the
lower left corner.The velocity of the cue ball is the same as it
travels around the pool table until it goes into a corner pocket.We
are going to hit the cue ball at a 45 angle.The angle of incidence
equals the angle of reflection.
