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Billiards and Snooker are some of the Billiards and Snooker are some of the most mathematical of games. most mathematical of games.
Why is that? Why is that?
This lesson explores some of the interesting patterns that are created when you track the path of a billiard ball across the table.
Learning Content:
Number strand: factors and ratios Space strand: angles, symmetry (reflections) Pattern and Algebra: generalisation Working Mathematically
Size of table Number of
bounces
Final pocket
6×10
4×5
6×8
9×15
8×10
99×100
M×N
see any patterns that might help you form theories or conjectures? Draw two more tables of any size that’s going to help you prove your theory.
Size of table Number ofbounces
Final pocket
6×10 6 Top right
4×5
6×8
9×15
8×10
99×100
M×N
As soon as you know the size of the table, can you predict:
- How many bounces will it make until it goes into a pocket?
- Which pocket will it finally go into?
Size of table Number ofbounces
Final pocket
6×10 6 Top right
4×5 7 Top left
6×8 5 Bottom right
9×15 6 Top right
8×10 7 Top right
99×100 197 Top
M×N M+N-2 Depends on whether M+N-2 is an odd or even
number
Summarizing Rules
1. Number of Bounces
Patterns such as 6 by 10, 9 by 15, or even 15 by 25, are all identical to 3 by 5. Hence ratio and common factors are important components of the rule.
which is: If M and N have no common factors then there are M + N - 2 bounces.
If M and N share a common factor of f, then the number of bounces is (M + N)/f - 2.
M
N
2. Final PocketThis depends on which direction the ball is facing after
M + N - 2 bounces (assuming M and N have no common factors. Each bounce changes the direction by 90 degrees. a) If M + N is even. An even number of 90 degree turns means the ball has
turned either 180 or 360 degrees, so it must be facing the direction it came from (impossible),
or towards the top right hand pocket.
Hence tables such as 3 by 5, 15 by 11, 6 by 10 will all finish in the top right hand corner. b) If M + N - 2 is odd then after an odd number of 90
degree turns the ball must finish in the top left or the bottom right corner.
