A king wants to choose his successor using a dice-rolling contest where the best total wins. But with the contestants rolling behind closed doors, the highest score may not be legitimate. Can you assess the results with mathematics and ensure that the new ruler will be an honest one?This lesson explores two different areas of mathematics: probability and number theory.Probability is the study of the likelihood of random events. If you roll one fair six-sided die, every face should land on top one-sixth of the time. If you assume every roll is independent, probability tells us that the chance of multiple events happening can be calculated by multiplying
the individual probabilities together.The reasoning behind Cassandra’s disqualification relies on the extremely low probability of rolling the single highest number 40 times. Ruling out possibilities based on their extreme unlikelihood is the basis of hypothesis testing
in statistics.Number theory is the study of integers and their properties. One of the first topics in most introductory number theory courses is modular arithmetic
, the idea that if we divide all integers by a particular number or “modulus,” there are addition and multiplication rules for the remainders.The reasoning behind Draco’s disqualification relies on the fact that 3 + 2 = 0 “modulo 5,” and so the sum of two numbers with remainders of 3 and 2 after dividing by 5 will always have a remainder of 0 after dividing by 5.If you’d like to explore modular arithmetic further, experiment with what happens if you divide a lot of square numbers (the result of multiplying integers by themselves) by 5. What remainders occur, and what remainders never seem to occur? Can you figure out why this happens?