Arithmetic billiards
In mathematics, arithmetic billiards provide a visual method to determine the least common multiple and the greatest common divisor of two whole numbers by making use of reflections inside a rectangle. This is an easy example of trajectory analysis of dynamical billiards.
Arithmetic billiards have been discussed as mathematical puzzles by Hugo Steinhaus and Martin Gardner.[1][2] Some teachers call it 'Paper Pool'.[3] They have been used as a source of questions in mathematical circles.[4]
How it works
changeThis method finds both the least common multiple, and greatest common divisor of two numbers. The largest number is drawn as a line with that length horizontally. The other length is drawn vertically.
Drawing a line at a 45 degree angle from the bottom right corner and every time the line hits a line it bounces off at the same 45 degree angle. This happens until it reaches a corner where it stops.
The total length of this line is the Lowest Common Multiple, whilst the distance between the bottom of the rectangle and the nearest change in angle is the lowest common multiple.[5]
References
change- ↑ Gardner, Martin (1984). Sixth Book of Mathematical Diversions from "Scientific American". University of Chicago Press. pp. 211–215. ISBN 0226282503.
- ↑ Steinhaus, Hugo (1999). Mathematical Snapshots (Dover Recreational Math Series ed.). Courier Corporation. p. 63. ISBN 0486409147.
- ↑ "Paper Pool Game". NCTM Illuminations. National Council of Teachers of Mathematics. Retrieved 10 January 2018.
- ↑ Tanton, James (2012). Mathematical Galore! The First Five Years of the St. Mark's Institute of Mathematics. The Mathematical Association of America. pp. 145–156. ISBN 978-0883857762.
- ↑ Perucca, Antonella (April 24, 2018). "Arithmetic Billiards". Plus Magazine. University of Cambridge. Retrieved December 23, 2018.