1. SHOOTING POOL Day Two Alvaro Francisco Manuel Western Oregon University June 3, 2015

2. INTRODUCTION  Review of Day One  Problem Statement  Strategy  Theorems  Figure out the hardest straight- in pool shot

3. FORMULAS  Theorem 1 If the gcd(n,m)= k, then B(n,m) = B (n/k, m/k) and P(n,m) = P (n/k, m/k).  Theorem 2 If the gcd(n,m) = 1 then B(n,m) = n + m − 2.  Theorem 3 Let T(n,m) be a given table with gcd(n, m) = 1. Then P(n,m) = UR n and m are both odd UL n is even and m is odd LR n is odd and m is even

4. PROBLEM STATEMENT  Find a formula for the number of bounces and the exit corner when first aimed at (1,1,1) from (0,0,0) in an (n,m,k) box.

5. THE SET UP  We are going to use the y-axis, x-axis, and z-axis  Starting position is at the (0,0,0)  (n,m,k) where n (y-axis), m (x-axis), and k (z-axis)  Following the path of the ball as it bounces the top and bottom, right and left, front and back.

6. ASSUMPTION

7. NOTATION  Our box with three dimensions as (n,m,k)  (n,m,k) box is defined to be the size of box.  B(n,m,k) is defined to be the number of bounces  C(n,m,k) is defined to be the corner pocket the ball lands in. Exit Corner 1Lower Left Front 2Lower Right Front 3Upper Left Front 4Upper Right Front 5Lower Left Back 6Lower Right Back 7Upper Left Back 8Upper Right Back 1 2 7 4 3 5 6 8

8. MY STRATEGY  Plan is to write down all the points the ball goes through.

9. EXAMPLE OF A (3,4,2) BOX 3 4 2 S.P1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th 12th 3 0123210123210 40123432101234 20121012101210 S.P1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th 12th 3 0123210123210 40123432101234 20121012101210 1 1 1 (1,1,1)

10. DATA Dimensions# bouncesExit Corner (3,3,3)0Upper Right Back (4,4,4)0Upper Right Back (2,1,1)1Upper Left Front (4,2,2)1Upper Left Front (6,3,3)1Upper Left Front (3,1,2)5Bottom Left Back (2,3,1)5Top left Front (1,2,3)5Bottom Right Front (2,1,3)5Top Left Front (3,2,1)5Bottom Left Front (1,3,2)5Bottom Left Back

11. FINDINGS  (n,n,n) box has zero bounces and ended up at the Upper Right Back exit corner  We can reduce large boxes with the gcd(n,m,k)  The permutations of (n,m,k) doesn’t change the number of bounces

12. THEOREMS

13.  Theorem 6: Suppose we have an (n,m,k) box with gcd(n,m,k) = 1, then the location of the exit corner can be found by Exit Corner =

14. S.P1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th 12th 3 0123210123210 40123432101234 20121012101210

15. EXAMPLE 3 4 2

16. THE HARDEST STRAIGHT – IN POOL SHOT  Here is our scenario

17.  What distance from the cue ball to the object ball makes the shot most difficult?

18.  How to measure the difficulty?

19. What distance from the cue ball to the object ball makes the shot most difficult?

23. SOLUTION

24. SUMMARY  Worked out lots of examples  Found formula  Learned more about pool

25. 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.  "Experimental Feature." Wolfram|Alpha: Computational Knowledge Engine. N.p., 3 June 2015. Web. 03 June 2015.  Marby, Rick (2010) The College Mathematics Journal, Vol. 41, No. 1 Mathematical Association of America. Web. http://www.jstor.org/stable/10.4169/89589048X345119

26. THANK YOU