PowerPoint Presentation

Shooting Pool

Academic ExcellenceAlvaro Francisco ManuelMay 28, 2015

From google drive on 4/7/151Problem StatementFind a general formula to predict the number of bounces a cue ball will have before landing in a corner pocket for any size pool table when shot at a 45 angle from the lower left corner of a pool table.Find a general formula to predict which corner pocket the cue ball will go in.

We have a (3,4) pool table.Starting position was at the lower left corner.Five bounces before landing in the lower right corner pocket. Assumption

https://www.google.com/search?q=cue+sticks+clip+art&rlz=1C1CHXU_enUS634US634&espv=2&biw=830&bih=928&tbm=isch&imgil=Z4nH0gzGajrHpM%253A%253BYIB8eaD1kKYtqM%253Bhttp%25253A%25252F%25252Fwww.fotosearch.com%25252Fillustration%25252Fcue-stick.html&source=iu&pf=m&fir=Z4nH0gzGajrHpM%253A%252CYIB8eaD1kKYtqM%252C_&usg=__ihMC_n0L9UORS1QDUbItHy-cS2A%3D&ved=0CDcQyjc&ei=u3sZVdWvEc-WoQTTsoLYAw#imgdii=wVXrFM4xeTGcQM%3A%3B6QXGKtEdY5acpM%3BwVXrFM4xeTGcQM%3A&imgrc=wVXrFM4xeTGcQM%253A%3Bff_ojDrFutcUYM%3Bhttp%253A%252F%252Flessonpix.com%252Fdrawings%252F6137%252F90x85%252FPool%252BCue.png%3Bhttp%253A%252F%252Flessonpix.com%252Fpictures%252F6135%252FPool%252520Cue%3B90%3B85

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.

5NotationWe 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.6My strategyThe plan was to work on many examples starting with small table size.TablesDimensions# of BouncesLocation1x10UR1x21LR1x32UR1x43LR1x54UR1x65LR2x11UL2x20UR2x33UL2x41LR2x55UL2x62UR2x77UL2x83LR2x99UL2x104UR3x12UR3x23LR3x30UR3x45LR3x56UR3x61LR3x78UR3x89LR3x92UR3x1011LR4x13UL4x21UL4x35UL4x40UR4x57UL4x63UL4x79ULDimensions# of BouncesLocation5x47LR5x50UR5x69LR5x710UR6x711UL6x85LR6x93UL6x106UR6x1115UL7x813LR7x914UR7x1015LR8x23UL8x39UL8x410UL8x511UL9x411LR9x512UR9x63LR9x714UR10x66UR10x715UL10x87LR10x917UL11x312UR11x413LR11x514UR11x615LRTablesDimensions# of BouncesLocation1x10UR1x21LR1x32UR1x43LR1x54UR1x65LR2x11UL2x20UR2x33UL2x41LR2x55UL2x62UR2x77UL2x83LR2x99UL2x104UR3x12UR3x23LR3x30UR3x45LR3x56UR3x61LR3x78UR3x89LR3x92UR3x1011LR4x13UL4x21UL4x35UL4x40UR4x57UL4x63UL4x79ULDimensions# of BouncesLocation5x47LR5x50UR5x69LR5x710UR6x711UL6x85LR6x93UL6x106UR6x1115UL7x813LR7x914UR7x1015LR8x23UL8x39UL8x410UL8x511UL9x411LR9x512UR9x63LR9x714UR10x66UR10x715UL10x87LR10x917UL11x312UR11x413LR11x514UR11x615LRTablesDimensions# of BouncesLocation1x21LR1x32UR1x43LR1x54UR1x65LR2x11UL2x20UR2x33UL2x41LR2x55UL2x62UR2x77UL2x83LR2x99UL2x104UR3x12UR3x23LR3x30UR3x45LR3x56UR3x61LR3x78UR3x89LR3x92UR3x1011LR4x13UL4x21UL4x35UL4x40UR4x57UL4x63UL4x79ULDimensions# of BouncesLocation5x50UR5x69LR5x710UR6x711UL6x85LR6x93UL6x106UR6x1115UL7x813LR7x914UR7x1015LR8x23UL8x39UL8x410UL8x511UL8x123UL9x512UR9x63LR9x714UR10x66UR10x715UL10x87LR10x917UL11x312UR11x413LR11x514UR11x615LRPatterns 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. Look at T(2,3) , T(4,6), and T(8,12), and B(1,2) = B(2,4) = B(4,8)= 3.In general, it seemed that B(n,m) = B(kn,km)FormulasDefinition A Step is when the cue ball moves through one square, it moves 1 unit in the vertical direction and one unit in the horizontal direction.

T(3,4)Should13Definition A Step is when the cue ball moves through one square, it moves 1 unit in the vertical direction and one unit in the horizontal direction.

Vertical Bounce is a bounce that occurs at the top or bottom edge of the pool table.Horizontal Bounce is a bounce that occurs at the right or left edge of the pool table.

Should14T(3,4)Vertical Bounce = 3 (n is the vertical dimension)T/B = 3,6,9 and goes in at 12

36912Theorem 2 If the gcd(n,m) = 1 then B(n,m) = n + m 2.

T(3,4)Vertical Bounce = 3 (n is the vertical dimension)T/B = 3,6,9 and goes in at 12

Horizontal Bounce = 2 (m is the horizontal dimension)L/R = 4,8 and goes in at 12

369481212Theorem 2 If the gcd(n,m) = 1 then B(n,m) = n + m 2.

Theorem 2 If the gcd(n,m) = 1 then B(n,m) = n + m 2.

lcm(3,4) = 12T/B : 3,6,9, and 3(4)=12Number of bounces is 3 = 4 -1= m -1L/R : 4,8, and 4(3)=12Number of bounces is 2 = 3-1 = n - 1 Total bounces: 5 = 3+2 = (4-1)+(3-1) = (m-1)+(n-1)= n+m-2.

4893612Summary solutionP(448,320) = P(448/64, 320/64) = P(7,5) = UR by Thm. 3

4 x 818Thank youReferencesStevenson, Fred (2013) Mathematics Explorations for Ages 10 to 100: A Travel Guide to Math Discovery. University of Arizona 2013.Doctors Lorenzo and Rachel, (1996). Bouncing Cue Ball. THE MATH FORUM. Retrieved from http://mathforum.org/library/drmath/view/54893.html.