Arithmetic billiards

In mathematics, arithmetic billiards is a way to show the least common multiple and the greatest common divisor of two whole numbers. It makes use of reflection

Arithmetic billiards
The arithmetic billiard for the numbers 15 and 40.
The arithmetic billiard for the numbers 15 and 40: the greatest common divisor is 5, the least common multiple is 120.

In mathematics, arithmetic billiards is a way to show the least common multiple and the greatest common divisor of two whole numbers. It makes use of reflections inside a rectangle.

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]

The arithmetic billiard for the numbers 10 and 40.
The arithmetic billiard for the numbers 10 and 40.

How it works

This 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

  1. ↑ Gardner, Martin (1984). Sixth Book of Mathematical Diversions from "Scientific American". University of Chicago Press. pp. 211–215. ISBN 0226282503.
  2. ↑ Steinhaus, Hugo (1999). Mathematical Snapshots (Dover Recreational Math Series ed.). Courier Corporation. p. 63. ISBN 0486409147.
  3. ↑ "Paper Pool Game". NCTM Illuminations. National Council of Teachers of Mathematics. Archived from the original on 10 February 2018. Retrieved 10 January 2018.
  4. ↑ 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.
  5. ↑ Perucca, Antonella (April 24, 2018). "Arithmetic Billiards". Plus Magazine. University of Cambridge. Retrieved December 23, 2018.

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