1. SHOOTING POOL Academic Excellence Alvaro Francisco Manuel May 28, 2015

2. PROBLEM STATEMENT  Find 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.

4. 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.

5. ASSUMPTION

6. NOTATION  We 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: LL  Lower Right: LR  Upper Left: UL  Upper Right: UR  T(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.

7. MY STRATEGY  The plan was to work on many examples starting with small table size.

8. TABLES Dimensions# of BouncesLocation 1x10UR 1x21LR 1x32UR 1x43LR 1x54UR 1x65LR 2x11UL 2x20UR 2x33UL 2x41LR 2x55UL 2x62UR 2x77UL 2x83LR 2x99UL 2x104UR 3x12UR 3x23LR 3x30UR 3x45LR 3x56UR 3x61LR 3x78UR 3x89LR 3x92UR 3x1011LR 4x13UL 4x21UL 4x35UL 4x40UR 4x57UL 4x63UL 4x79UL Dimensions# of BouncesLocation 5x47LR 5x50UR 5x69LR 5x710UR 6x711UL 6x85LR 6x93UL 6x106UR 6x1115UL 7x813LR 7x914UR 7x1015LR 8x23UL 8x39UL 8x410UL 8x511UL 9x411LR 9x512UR 9x63LR 9x714UR 10x66UR 10x715UL 10x87LR 10x917UL 11x312UR 11x413LR 11x514UR 11x615LR

9. TABLES Dimensions# of BouncesLocation 1x10UR 1x21LR 1x32UR 1x43LR 1x54UR 1x65LR 2x11UL 2x20UR 2x33UL 2x41LR 2x55UL 2x62UR 2x77UL 2x83LR 2x99UL 2x104UR 3x12UR 3x23LR 3x30UR 3x45LR 3x56UR 3x61LR 3x78UR 3x89LR 3x92UR 3x1011LR 4x13UL 4x21UL 4x35UL 4x40UR 4x57UL 4x63UL 4x79UL Dimensions# of BouncesLocation 5x47LR 5x50UR 5x69LR 5x710UR 6x711UL 6x85LR 6x93UL 6x106UR 6x1115UL 7x813LR 7x914UR 7x1015LR 8x23UL 8x39UL 8x410UL 8x511UL 9x411LR 9x512UR 9x63LR 9x714UR 10x66UR 10x715UL 10x87LR 10x917UL 11x312UR 11x413LR 11x514UR 11x615LR

10. TABLES Dimensions# of BouncesLocation 1x21LR 1x32UR 1x43LR 1x54UR 1x65LR 2x11UL 2x20UR 2x33UL 2x41LR 2x55UL 2x62UR 2x77UL 2x83LR 2x99UL 2x104UR 3x12UR 3x23LR 3x30UR 3x45LR 3x56UR 3x61LR 3x78UR 3x89LR 3x92UR 3x1011LR 4x13UL 4x21UL 4x35UL 4x40UR 4x57UL 4x63UL 4x79UL Dimensions# of BouncesLocation 5x50UR 5x69LR 5x710UR 6x711UL 6x85LR 6x93UL 6x106UR 6x1115UL 7x813LR 7x914UR 7x1015LR 8x23UL 8x39UL 8x410UL 8x511UL 8x123UL 9x512UR 9x63LR 9x714UR 10x66UR 10x715UL 10x87LR 10x917UL 11x312UR 11x413LR 11x514UR 11x615LR

11. PATTERNS I FOUND  Any 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)

12. FORMULAS

13. DEFINITION  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)

14. DEFINITION  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.

15.  T(3,4)  Vertical Bounce = 3 (n is the vertical dimension)  T/B = 3,6,9 and goes in at 12 3 6 9 12 Theorem 2 If the gcd(n,m) = 1 then B(n,m) = n + m − 2.

16.  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 3 6 9 4 8 12 Theorem 2 If the gcd(n,m) = 1 then B(n,m) = n + m − 2.

17.  lcm(3,4) = 12  T/B : 3,6,9, and 3(4)=12  Number of bounces is 3 = 4 -1= m -1  L/R : 4,8, and 4(3)=12  Number of bounces is 2 = 3-1 = n - 1  Total bounces: 5 = 3+2 = (4-1)+(3-1) = (m-1)+(n-1) = n+m-2. 4 8 93 6 12

18. SUMMARY SOLUTION  P(448,320) = P(448/64, 320/64) = P(7,5) = UR by Thm. 3

19. THANK YOU

20. REFERENCES  Stevenson, 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.